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A scaling perspective on the distribution of executive compensation
Physica A xxx (xxxx) xxx
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A scaling perspective on the distribution of executive compensation ∗
Thitithep Sitthiyot a , , Pornanong Budsaratragoon a , Kanyarat Holasut b a
Department of Banking and Finance, Faculty of Commerce and Accountancy, Chulalongkorn University, Mahitaladhibesra Bld., 10th Fl., Phayathai Rd., Pathumwan, Bangkok 10330, Thailand b Department of Chemical Engineering, Faculty of Engineering, Khon Kaen University, Mittapap Rd., Muang District, Khon Kaen 40002, Thailand
article
info
Article history: Received 9 November 2018 Received in revised form 23 October 2019 Available online xxxx Keywords: Scale invariance Self-similarity Executive compensation distribution Lorenz curve
a b s t r a c t We investigate scale invariance or self-similarity in the distribution of average executive compensation defined as total executive compensation for each company divided by the number of executives in that company. Using annual data on companies listed in the Stock Exchange of Thailand between 2002 and 2015, the average executive compensation is categorized into three groups according to time period, industry type, and company size. The results from estimating the Lorenz curve and the Kolmogorov– Smirnov test indicate that the distributions of average executive compensation are statistically scale invariance or self-similar across time period, industry type, and company size with p-values greater than 0.01 in all cases. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The search for statistical regularities in income and wealth distributions has a long history. Pareto [1,2] observes that the distribution of income follows a power law and varies very little in space and time. Different people and different eras yield very similar results. Pareto [2] also notes that the shape of income distribution is remarkably stable. Mandelbrot [3] observes that, over a certain range of values of income, its distribution is not markedly influenced either by socioeconomic structure of the community under study or by definition of income being chosen. Klass et al. [4] examine wealth distribution of the 400 richest published by the Forbes magazine and find that the wealth of these people is distributed power law. Klass et al. [4] note that although the richest made their fortunes in various different ways, the distribution of their wealth exhibits a property of scale invariance. Drawing the idea from Anderson [5], Klass et al. [4] argue that the same dynamical rules of gains and losses apply across the entire economy independently of particular sector or wealth and sophistication of different individuals. Chakrabarti et al. [6] review literatures on income and wealth distributions and conclude that income and wealth distributions follow a universal pattern irrespective of differences in culture, history, social structure, indicators of prosperity, and economic policies adopted in different countries. For more than a century, numerous studies have investigated and confirmed that the income and wealth distributions obey power laws which imply that they have no characteristic scale (see Jagielski et al. [7] and references therein). According to Bak [8], West and Brown [9], and West [10], scale invariance is a property inherent to power laws. It naturally reflects underlying generic features and physical principles that are independent of detailed dynamics or specific characteristics of particular systems. According to Chakrabarti et al. [6], there tends to be an agreement among scholars in ∗ Corresponding author. E-mail addresses: [email protected] (T. Sitthiyot), [email protected] (P. Budsaratragoon), [email protected] (K. Holasut). https://doi.org/10.1016/j.physa.2019.123556 0378-4371/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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T. Sitthiyot, P. Budsaratragoon and K. Holasut / Physica A xxx (xxxx) xxx
economics, statistics, and physics that the upper tail of income distribution can be well described by power law. However, for the lower part of the distribution, it is still debatable whether income distribution follows gamma or log-normal, both of which also imply that income distribution is scale invariance. In addition, recent study by Broido and Clauset [11] uses 927 network data sets to examine the universality of scale invariance in biological, informational, social, technological, and transportation networks by fitting power law model, testing its statistical property, and comparing it with alternative distributions. They find that there are significant differences in degree of scale invariance among these networks. Empirical results found in Broido and Clauset [11] indicate that scale invariance may not be uniformly distributed across different types of networks as claimed by Barabási [12]. While power law, gamma, and log-normal distributions imply that data have a property of scale invariance, we introduce an alternative method that is simple and can be used to test scale invariance or self-similarity in any types of data, irrespective of their distributions. In our study, we use annual data on average executive compensation of companies listed in the Stock Exchange of Thailand between 2002 and 2015 as a case study in order to demonstrate our method. Average executive compensation is defined as the total executive compensation for each company divided by the number of executives in that company. We use the average executive compensation because the Stock Exchange of Thailand does not collect data on executive compensation for each individual executive. Therefore, the average executive compensation for a given company in a given year is used as a proxy for individual executive compensation for that company in that year. Our dataset contains almost 6000 observations. The average executive compensation data are sorted into three groups based on time period, industry type, and company size. We would like to investigate whether or not the distributions of average executive compensation vary across different dimensions, namely, time period, type of industry, and company size. If the distributions of average executive compensation are scale invariance, then they should look statistically selfsimilar across all three different types of data grouping. As recommended by Brock [13], knowing nature of the distribution of executive compensation is useful because it imposes discipline on theory formation in that such a theory must generate predictions that are consistent with the observed scaling features. 2. Methods and data 2.1. Methods In this study, scale invariance is defined as self-similarity of distributions of average executive compensation across three types of data grouping which are time period (T), industry type (I), and company size (C). Let y (x) = the Lorenz function, where y = cumulative normalized average executive compensation and x = cumulative normalized rank of companies. If the distributions of average executive compensation are scale invariance or yT (x) ∼ = yI (x) ∼ = yC (x) , given 0 ≤ x ≤ 1, then the proportions of companies that fall in the same bin on the cumulative normalized rank of companies axis (the x-axis) should look statistically self-similar across time period (T), industry type (I), and company size (C). In order to test this hypothesis, we begin our method by sorting the average executive compensation data into three groups according to time period, industry type, and company size. The sorted data for each group are then normalized and arranged in an ascending order. For each type of data classifications, we plot Cartesian coordinates where the abscissa is the cumulative normalized rank of companies (x) and the ordinate is the cumulative normalized average executive compensation (y). This would give us the actual Lorenz plots for three different types of data grouping according to time period, industry type, and company size. Next, we try to fit a representative Lorenz curve to our actual Lorenz plots for each type of data groupings. The use of a representative Lorenz curve for each type of data groupings is based on the condition that the shape of the distribution of average executive compensation is relatively stable across time as empirically found in income distribution by Pareto [2], in firm sizes distribution by Axtell [14], and in wealth distribution by Klass et al. [4]. However, in order to test whether this condition is practically usable, we have to demonstrate that, for each type of data groupings, the Lorenz plots based on actual observations are statistically stable across time period. This could be done by dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins and counting the actual proportions of companies that fall in each corresponding bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). After obtaining the proportions of companies that fall in each corresponding bin (total of twenty bins) for all actual Lorenz plots, each of which representing each time period, we then conduct the pairwise Kolmogorov–Smirnov test to compare whether the proportions of companies that fall in the same bin (total of twenty bins) are statistically different from each other across time period or not. If the results from the pairwise Kolmogorov–Smirnov test indicate that, for each type of data groupings, the actual Lorenz plots are statistically not different from each other, we could proceed by pooling the data across time period and fitting a representative Lorenz curve to the actual Lorenz plots according to each type of data groupings. Note that we choose the Kolmogorov–Smirnov test because it is commonly used to determine whether or not two datasets differ statistically. Its advantage is that it makes no assumption about the distribution of the data. While there are many nonlinear functions that could be used to fit the Lorenz curve as discussed in Chakrabarti et al. [6], Drăgulescu and Yakovenko [15], and Fellman [16], to keep our method simple and realistic, we use the polynomial function to estimate the Lorenz curve on the conditions that the estimated equation has to be an increasing function and must pass two coordinates which are (0, 0) and (1, 1). This curve fitting method based on minimizing error sum of Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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squares is applied for each type of data groupings according to time period (T), industry type (I), and company size (C). This would give us three equations for the estimated Lorenz curves, each of which represents the distribution of average executive compensation for three different types of data grouping. By dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins and utilizing the coefficients obtained from the three estimated equations for the Lorenz curves, for a given value of cumulative normalized average executive compensation (y), we can work backward to calculate the value of cumulative normalized rank of companies (x) in order to find out the proportion of the listed companies that falls in each corresponding bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). Finally, we conduct the Kolmogorov–Smirnov test to compare whether or not the proportions of companies that fall in the same bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis) calculated from three different types of data grouping are statistically different from each other. Not rejecting the null hypothesis, that the proportions of companies falling in the same bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis) are statistically self-similar across three types of data grouping, would imply that yT (x) ∼ = yI (x) ∼ = yC (x). 2.2. Data The data used in this study have been obtained from the Stock Exchange of Thailand which has compiled the data on total executive compensations and total number of executives from annual registration statements (Form 56-1) submitted by each of its listed companies every year since 2001 (except in 2008 where no data are available) as well as the data from the listed companies’ financial statements. According to the Stock Exchange of Thailand, executives are those who take positions of managing director, deputy managing directors, assistant managing directors, and accounting and finance manager. Compensations include salary, bonus, provident fund, meeting fee, warrant employee joint investment program, warrant employee stock option, as well as other remunerations, for example, car allowance, home rental, and insurance fee. The Stock Exchange of Thailand classifies industry into nine categories. They are agro and food industries, consumer products, finance, industries, property and construction, resources, services, technology, and others. The data from financial statements of the listed companies comprise items such as total assets, total sales, net profits, total revenues, and market capitalization. In our study, we use total assets as a representative for company sizes. By taking natural logarithm of company’s total assets, company size is then classified into ten groups. Due to the availability of the data, our study covers periods between 2002 and 2015. The data on average executive compensation categorized by time period and industry type consist of 5922 observations whereas those grouped by total assets as a proxy for size of company contain 5089 observations. Descriptive statistics for three types of average executive compensation data grouping and for data on companies’ total assets are provided in Table 1. Table 1 Descriptive statistics of average executive compensation and of total assets of companies between 2002 and 2015. The average executive compensation data are categorized based on time period, industry type, and company size. Total assets of companies are used as a representative of company size. Unit: Million Baht Data
Mean
Median
Mode
Maximum
Minimum
Standard deviation
Number of observations
Average executive compensation (time period) Average executive compensation (industry type) Average executive compensation (company size) Total assets of companies
3.880 3.880 3.972 39.425
2.621 2.621 2.662 3.006
0.200 0.200 1.044 6.109
49.198 49.198 49.198 2835.852
0.003 0.003 0.003 0.034
4.342 4.342 4.424 205.657
5922 5922 5089 5089
3. Results Recall that before fitting a representative Lorenz curve to the actual Lorenz plots for three different types of data groupings, namely, time period (T), industry type (I), and company size (C), it is important that we have to test whether or not the distribution of average executive compensation is statistically stable across years so that we can pool the annual data and fit a representative Lorenz curve to the actual Lorenz plots for each type of data groupings. Due to the number of observations for time period (T) and industry type (I) is identical which is equal to 5922, just being categorized in different ways, we can simultaneously test whether or not the actual Lorenz plots that represent the distribution of average executive compensation are statistically different across years for these two types of data categorization. For company size (C), we have to do a separate test since the number of observations is equal to 5089. Tables 2 and 3 report values of D-statistic from the pairwise Kolmogorov–Smirnov test. They indicate that, for each type of data groupings, namely, time period (T), industry type (I), and company size (C), all actual Lorenz plots are not statistically different from each other across years (total of thirteen years) with values of D-statistic between 0.00 and 0.20 and p-values greater than 0.01 in all cases (see Tables S1 and S2 in Supplementary Materials for proportions of companies (x) that fall in the same bins across years (2002–15) grouped by time period (T), industry type (I), and company size (C)). Given these results, we can proceed to pool the data across years and try to fit a representative Lorenz curve to the actual Lorenz plots for each type of data groupings. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Table 2 Values of D-statistic from the pairwise Kolmogorov–Smirnov test show that, for time period (T) and industry type (I), there are no statistical differences in the distribution of average executive compensation across years with p-values, shown in parentheses, greater than 0.01 in all cases.
2002
2002
2003
2004
2005
2006
2007
2009
2010
2011
2012
2013
2014
2015
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.05 (1.000)
0.05 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.05 (1.000)
0.15 (0.965)
0.10 (1.000)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.05 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
2003 2004 2005 2006 2007 2009 2010 2011 2012 2013 2014
0.00 (1.000)
2015
Table 3 Values of D-statistic from the pairwise Kolmogorov–Smirnov test show that, for company size (C), there are no statistical differences in the distribution of average executive compensation across years with p-values, shown in parentheses, greater than 0.01 in all cases.
2002 2003 2004 2005 2006 2007 2009 2010 2011 2012 2013 2014 2015
2002
2003
2004
2005
2006
2007
2009
2010
2011
2012
2013
2014
2015
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.20 (0.771)
0.15 (0.965)
0.20 (0.771)
0.20 (0.771)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.20 (0.771)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.20 (0.771)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.05 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.00 (1.000)
0.05 (1.000)
0.05 (1.000)
0.00 (1.000)
0.05 (1.000) 0.00 (1.000)
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Figs. 1–3 show actual Lorenz plots of the distributions of average executive compensation for three different types of data grouping which are time period (T), industry type (I), and size of company (C) between 2002 and 2015 except 2008 (total of thirteen years). For time period (T), there are thirteen Lorenz plots, one for each year (Fig. 1). For industry type (I), the total number of Lorenz plots is one hundred and seventeen (Fig. 2). This number comes from nine industries, each of which has its own Lorenz plot for each year. For company size (C), there are one hundred and twenty-five Lorenz plots (Fig. 3) since we classify company size into ten groups. Each group has its own Lorenz plots for each year. However, there are five groups, each of which comprises only one company. Therefore, we do not count these observations as the Lorenz plot. This would give us one hundred and twenty-five Lorenz plots for company size (C). The actual Lorenz plots illustrated in Figs. 1–3 indicate that the distributions of average executive compensation classified by time period (T) are quite stable during the periods of study relative to those categorized by industry type (I) and company size (C) which show higher variations. Note that a conventional method used to estimate the Lorenz curve could be done by fitting a nonlinear function to the actual observations. However, it is possible that some nonlinear functions may result in the fitted Lorenz curve that is not realistic in a sense that not only does it not pass the coordinate (1, 1) but also gives us a negative value of cumulative normalized average executive compensation (y) for a given value of cumulative normalized rank of companies (x). As described in Section 2.1, there are many nonlinear techniques that could be used to fit the Lorenz curve to the actual observations. However, to keep our method simple and still realistic, we use polynomial function to estimate the Lorenz curve on the conditions that the estimated equation has to be an increasing function and must pass two coordinates which are (0, 0) and (1, 1). We find that the estimated equations for the Lorenz curve for three different types of data grouping could be described by the sixth degree polynomial functions. They are as follows: Time period :
yT = 0.5139x6 + 0.2017x2 + 0.2845x, R2 = 0.9921
(1)
Industry type :
y = 0.4564x + 0.2696x + 0.2740x, R = 0.9770
(2)
Company size :
y = 0.2999x + 0.3423x + 0.3578x, R = 0.9881
(3)
I
C
6
6
2
2
2
2
The estimated Lorenz curves are also illustrated in Figs. 1–3. Note that the estimated Lorenz curve for time period (yT ) fits the actual Lorenz plots reasonably well with R2 = 0.9921 while those for industry type (yI ) and company size (yC ) show higher variances with R2 = 0.9770 and 0.9881, respectively. Based on the three fitted Lorenz curves, we can calculate the Gini coefficient to measure inequality in the distribution of average executive compensation for three types of data grouping. We find that the Gini coefficients for time period, industry type, and company size are equal to 0.4364, 0.4159, and 0.3283, respectively, indicating that inequality of the distribution of average executive compensation is quite equal across different company size relative to across time period and industry type. Our next task is to use the estimated coefficients obtained from Eqs. (1)–(3) above to calculate the proportion of companies on the cumulative normalized rank of companies axis (the x-axis) that corresponds to each bin (total of twenty bins) on the cumulative normalized average executive compensation axis (the y-axis) according to three different types of data grouping. We obtain the proportion of companies that corresponds to each cumulative normalized average executive compensation bin by dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins (0.00–0.05, 0.06–0.10, 0.11–0.15,. . . , 0.96–1.00). For a given value of cumulative normalized average executive compensation (y), we can use the coefficients from each of the three estimated equations for the Lorenz curve to compute the value of cumulative normalized rank of companies (x), and find out the proportions of companies that fall in each bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). In order to test the property of scale invariance or self-similarity in the distributions of average executive compensation, we perform the Kolmogorov–Smirnov test to compare whether or not the proportions of companies that fall in the same bin on the cumulative normalized rank of companies axis (the x-axis) according to the three different types of data grouping are statistically different from one another. Our findings are reported in Table 4. The numerical findings from Table 4 indicate that, for each of the cumulative normalized average executive compensation bins, the proportions of companies grouped by time period, industry type, and company size do not differ significantly from each other. Among three types of data groupings, the largest difference is 0.0332 (bin 1) while the smallest difference is 0.0001 (bin 1 and bin 6). The Kolmogorov–Smirnov test results, as shown in Table 4, confirm our numerical findings in that the difference between the distribution of proportions of companies categorized by time period vs. by industry type is statistically insignificant with p-value = 1.000. The difference between the distribution of proportions of companies grouped by time period vs. by company size is also statistically insignificant with p-value = 0.497. In addition, the difference between the distribution of proportions of companies sorted by industry type vs. by company size is statistically insignificant with p-value = 0.771. The results from the Kolmogorov–Smirnov test indicate that there are no statistical differences among the distributions of proportions of companies according to three different types of data grouping for p-values are greater than 0.01 in all cases. Our empirical findings suggest that the distributions of average executive compensation are scale invariance or self-similar. That is time period, type of industry, and company size have no effects on the distributions of average executive compensation. Our results are consistent with what found in previous studies as discussed in Section 1. However, those studies implicate this statistical regularity from fitting power law, gamma, or log-normal distributions to the actual income and wealth observations. We use an alternative method but come up with similar conclusion. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Fig. 1–3. Actual Lorenz plots of the distribution of average executive compensation and estimated Lorenz curves for three different types of data grouping. Cumulative normalized average executive compensation (y) plotted against cumulative normalized rank of companies (x) for three different types of data grouping according to time period (Fig. 1), industry type (Fig. 2), and company size (Fig. 3) between 2002 and 2015. Given the conditions that the fitted Lorenz curve has to be an increasing function and must pass (0, 0) and (1, 1), the estimated equations for the Lorenz curve for each type of data groupings can be described by the sixth degree polynomial functions. Note that the estimated Lorenz curve for time period fits the actual observations reasonably well while those for industry type and company size show higher variances.
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Table 4 Proportions of companies that are in the same bins (total of twenty bins) for three different types of data grouping and results from the KolmogorovSmirnov test. Empirical results show that the distributions of the proportion of companies based on three different types of data grouping are not statistically different from each other with p-values greater than 0.01 in all three cases. Note that the results do not change when the number of bins is ten or forty. Cumulative normalized average executive compensation
0.00–0.05 0.06–0.10 0.11–0.15 0.16–0.20 0.21–0.25 0.26–0.30 0.31–0.35 0.36–0.40 0.41–0.45 0.46–0.50 0.51–0.55 0.56–0.60 0.61–0.65 0.66–0.70 0.71–0.75 0.76–0.80 0.81–0.85 0.86–0.90 0.91–0.95 0.96–1.00
Proportions of companies based on 3 different data groupings
Kolmogorov–Smirnov test results (between 2 groups)
Time period
Industry type
Company size
Time period vs. industry type
Time period vs. company size
Industry type vs. company size
0.1580 0.1326 0.1133 0.0950 0.0778 0.0634 0.0521 0.0436 0.0372 0.0323 0.0285 0.0254 0.0229 0.0209 0.0192 0.0177 0.0165 0.0154 0.0145 0.0136
0.1579 0.1265 0.1066 0.0903 0.0759 0.0635 0.0534 0.0454 0.0391 0.0342 0.0302 0.0271 0.0244 0.0223 0.0205 0.0189 0.0176 0.0164 0.0154 0.0145
0.1248 0.1043 0.0911 0.0811 0.0728 0.0653 0.0585 0.0523 0.0468 0.0419 0.0377 0.0341 0.0309 0.0283 0.0259 0.0239 0.0222 0.0207 0.0193 0.0181
D-statistic = 0.05 p-value = 1.000
D-statistic = 0.25 p-value = 0.497
D-statistic = 0.20 p-value = 0.771
4. Conclusions and remarks While studies about scale invariance or self-similarity in income and wealth distributions have been generally associated with testing whether or not such distributions follow power law, gamma, or log-normal, in this study, we present an alternative method for testing scale invariance or self-similarity. Our method is simple and can be employed to test scale invariance or self-similarity in any types of data, regardless of their distributions. To demonstrate our method, we use the average executive compensation of companies listed in the Stock Exchange of Thailand between 2002 and 2015 as a case study. Despite the complexity of interaction and/or interconnection among companies of different sizes, operating in different industries over a long period of time, each of which at some point may change or adjust its own compensation scheme, as well as a number of major events such as severe flooding, economic recession and recovery, and political instability in Thailand during that period, the empirical evidence on scale invariance in the distribution of average executive compensation found in our study indicates that the overall distributions remain statistically unchanged across time period, industry type, and company size. As argued by West and Brown [9], scale invariance naturally reflects the underlying generic features independently of detailed characteristics of particular systems. Therefore, any theory developed in order to explain the distribution of executive compensation should predict the statistical regularity of scale invariance as previously recommended by Brock [13]. Reference data All data analyzed during this study are included in this article. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Thitithep Sitthiyot: Conceptualization, Methodology, Formal analysis, Writing — original draft, Writing — Review&Editing. Pornanong Budsaratragoon: Investigation, Resources, Writing — Review&Editing. Kanyarat Holasut: Validation, Writing — Review&Editing. Acknowledgments The authors appreciate all reviewers for their useful comments and suggestions. Thitithep Sitthiyot is grateful to Dr. Suradit Holasut for invaluable guidance and fruitful comments. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physa.2019.123556. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
V. Pareto, Cours D’économie Politique, Vol. 2, Macmillan, London, 1897. V. Pareto, Manual of Political Economy, Augustus M. Kelley, New York, 1927, Schwier, A. S. (trans.), B. Mandelbrot, The Pareto-Lévy law and the distribution of income, Internat. Econom. Rev. 1 (2) (1960) 79–106. O.S. Klass, O. Biham, M. Levy, O. Malcai, S. Solomon, The forbes 400 and the pareto wealth distribution, Econom. Lett. 90 (2006) 290–295. P.W. Anderson, Some thoughts about distribution in economics, in: W.B. Arthur, S.N. Durlauf, D.A. Lane (Eds.), The Economy as an Evolving Complex System II, SFI Studies in the Sciences of Complexity, Vol. XXVII, Westview Press, Oxford, 1997, pp. 565–566. B.K. Chakrabarti, A. Chakraborti, S.R. Chakravarty, A. Chatterjee, Econophysics of Income and Wealth Distributions, Cambridge University Press, New York, 2013. M. Jagielski, K. Czyzewski, R. Kutner, H.E. Stanley, Income and wealth distribution of the richest Norwegian individuals: An inequality analysis, Physica A 474 (2017) 330–333. P. Bak, How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York, 1999. G.B. West, J.H. Brown, Life’s universal scaling laws, Phys. Today 57 (9) (2004) 36–42. G.B. West, Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies, Penguin Press, New York, 2017. A.D. Broido, A. Clauset, Scale-free networks are rare, 2018, unpublished paper at: https://arxiv.org/abs/1801.03400v1. A.-L. Barabási, Network Science, Cambridge University Press, Cambridge, 2016. W.A. Brock, Scaling in economics: A reader’s guide, Ind. Corp. Change 8 (3) (1999) 409–446. R.L. Axtell, Zipf distribution of U.S. firm sizes, Science 293 (2001) 1818–1820. A. Drăgulescu, V.M. Yakovenko, Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States, Physica A 299 (2001) 213–221. J. Fellman, Regression analyses of income inequality indices, Theor. Econ. Lett. 8 (2018) 1793–1802.
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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A scaling perspective on the distribution of executive compensation ∗
Thitithep Sitthiyot a , , Pornanong Budsaratragoon a , Kanyarat Holasut b a
Department of Banking and Finance, Faculty of Commerce and Accountancy, Chulalongkorn University, Mahitaladhibesra Bld., 10th Fl., Phayathai Rd., Pathumwan, Bangkok 10330, Thailand b Department of Chemical Engineering, Faculty of Engineering, Khon Kaen University, Mittapap Rd., Muang District, Khon Kaen 40002, Thailand
article
info
Article history: Received 9 November 2018 Received in revised form 23 October 2019 Available online xxxx Keywords: Scale invariance Self-similarity Executive compensation distribution Lorenz curve
a b s t r a c t We investigate scale invariance or self-similarity in the distribution of average executive compensation defined as total executive compensation for each company divided by the number of executives in that company. Using annual data on companies listed in the Stock Exchange of Thailand between 2002 and 2015, the average executive compensation is categorized into three groups according to time period, industry type, and company size. The results from estimating the Lorenz curve and the Kolmogorov– Smirnov test indicate that the distributions of average executive compensation are statistically scale invariance or self-similar across time period, industry type, and company size with p-values greater than 0.01 in all cases. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The search for statistical regularities in income and wealth distributions has a long history. Pareto [1,2] observes that the distribution of income follows a power law and varies very little in space and time. Different people and different eras yield very similar results. Pareto [2] also notes that the shape of income distribution is remarkably stable. Mandelbrot [3] observes that, over a certain range of values of income, its distribution is not markedly influenced either by socioeconomic structure of the community under study or by definition of income being chosen. Klass et al. [4] examine wealth distribution of the 400 richest published by the Forbes magazine and find that the wealth of these people is distributed power law. Klass et al. [4] note that although the richest made their fortunes in various different ways, the distribution of their wealth exhibits a property of scale invariance. Drawing the idea from Anderson [5], Klass et al. [4] argue that the same dynamical rules of gains and losses apply across the entire economy independently of particular sector or wealth and sophistication of different individuals. Chakrabarti et al. [6] review literatures on income and wealth distributions and conclude that income and wealth distributions follow a universal pattern irrespective of differences in culture, history, social structure, indicators of prosperity, and economic policies adopted in different countries. For more than a century, numerous studies have investigated and confirmed that the income and wealth distributions obey power laws which imply that they have no characteristic scale (see Jagielski et al. [7] and references therein). According to Bak [8], West and Brown [9], and West [10], scale invariance is a property inherent to power laws. It naturally reflects underlying generic features and physical principles that are independent of detailed dynamics or specific characteristics of particular systems. According to Chakrabarti et al. [6], there tends to be an agreement among scholars in ∗ Corresponding author. E-mail addresses: [email protected] (T. Sitthiyot), [email protected] (P. Budsaratragoon), [email protected] (K. Holasut). https://doi.org/10.1016/j.physa.2019.123556 0378-4371/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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economics, statistics, and physics that the upper tail of income distribution can be well described by power law. However, for the lower part of the distribution, it is still debatable whether income distribution follows gamma or log-normal, both of which also imply that income distribution is scale invariance. In addition, recent study by Broido and Clauset [11] uses 927 network data sets to examine the universality of scale invariance in biological, informational, social, technological, and transportation networks by fitting power law model, testing its statistical property, and comparing it with alternative distributions. They find that there are significant differences in degree of scale invariance among these networks. Empirical results found in Broido and Clauset [11] indicate that scale invariance may not be uniformly distributed across different types of networks as claimed by Barabási [12]. While power law, gamma, and log-normal distributions imply that data have a property of scale invariance, we introduce an alternative method that is simple and can be used to test scale invariance or self-similarity in any types of data, irrespective of their distributions. In our study, we use annual data on average executive compensation of companies listed in the Stock Exchange of Thailand between 2002 and 2015 as a case study in order to demonstrate our method. Average executive compensation is defined as the total executive compensation for each company divided by the number of executives in that company. We use the average executive compensation because the Stock Exchange of Thailand does not collect data on executive compensation for each individual executive. Therefore, the average executive compensation for a given company in a given year is used as a proxy for individual executive compensation for that company in that year. Our dataset contains almost 6000 observations. The average executive compensation data are sorted into three groups based on time period, industry type, and company size. We would like to investigate whether or not the distributions of average executive compensation vary across different dimensions, namely, time period, type of industry, and company size. If the distributions of average executive compensation are scale invariance, then they should look statistically selfsimilar across all three different types of data grouping. As recommended by Brock [13], knowing nature of the distribution of executive compensation is useful because it imposes discipline on theory formation in that such a theory must generate predictions that are consistent with the observed scaling features. 2. Methods and data 2.1. Methods In this study, scale invariance is defined as self-similarity of distributions of average executive compensation across three types of data grouping which are time period (T), industry type (I), and company size (C). Let y (x) = the Lorenz function, where y = cumulative normalized average executive compensation and x = cumulative normalized rank of companies. If the distributions of average executive compensation are scale invariance or yT (x) ∼ = yI (x) ∼ = yC (x) , given 0 ≤ x ≤ 1, then the proportions of companies that fall in the same bin on the cumulative normalized rank of companies axis (the x-axis) should look statistically self-similar across time period (T), industry type (I), and company size (C). In order to test this hypothesis, we begin our method by sorting the average executive compensation data into three groups according to time period, industry type, and company size. The sorted data for each group are then normalized and arranged in an ascending order. For each type of data classifications, we plot Cartesian coordinates where the abscissa is the cumulative normalized rank of companies (x) and the ordinate is the cumulative normalized average executive compensation (y). This would give us the actual Lorenz plots for three different types of data grouping according to time period, industry type, and company size. Next, we try to fit a representative Lorenz curve to our actual Lorenz plots for each type of data groupings. The use of a representative Lorenz curve for each type of data groupings is based on the condition that the shape of the distribution of average executive compensation is relatively stable across time as empirically found in income distribution by Pareto [2], in firm sizes distribution by Axtell [14], and in wealth distribution by Klass et al. [4]. However, in order to test whether this condition is practically usable, we have to demonstrate that, for each type of data groupings, the Lorenz plots based on actual observations are statistically stable across time period. This could be done by dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins and counting the actual proportions of companies that fall in each corresponding bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). After obtaining the proportions of companies that fall in each corresponding bin (total of twenty bins) for all actual Lorenz plots, each of which representing each time period, we then conduct the pairwise Kolmogorov–Smirnov test to compare whether the proportions of companies that fall in the same bin (total of twenty bins) are statistically different from each other across time period or not. If the results from the pairwise Kolmogorov–Smirnov test indicate that, for each type of data groupings, the actual Lorenz plots are statistically not different from each other, we could proceed by pooling the data across time period and fitting a representative Lorenz curve to the actual Lorenz plots according to each type of data groupings. Note that we choose the Kolmogorov–Smirnov test because it is commonly used to determine whether or not two datasets differ statistically. Its advantage is that it makes no assumption about the distribution of the data. While there are many nonlinear functions that could be used to fit the Lorenz curve as discussed in Chakrabarti et al. [6], Drăgulescu and Yakovenko [15], and Fellman [16], to keep our method simple and realistic, we use the polynomial function to estimate the Lorenz curve on the conditions that the estimated equation has to be an increasing function and must pass two coordinates which are (0, 0) and (1, 1). This curve fitting method based on minimizing error sum of Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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squares is applied for each type of data groupings according to time period (T), industry type (I), and company size (C). This would give us three equations for the estimated Lorenz curves, each of which represents the distribution of average executive compensation for three different types of data grouping. By dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins and utilizing the coefficients obtained from the three estimated equations for the Lorenz curves, for a given value of cumulative normalized average executive compensation (y), we can work backward to calculate the value of cumulative normalized rank of companies (x) in order to find out the proportion of the listed companies that falls in each corresponding bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). Finally, we conduct the Kolmogorov–Smirnov test to compare whether or not the proportions of companies that fall in the same bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis) calculated from three different types of data grouping are statistically different from each other. Not rejecting the null hypothesis, that the proportions of companies falling in the same bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis) are statistically self-similar across three types of data grouping, would imply that yT (x) ∼ = yI (x) ∼ = yC (x). 2.2. Data The data used in this study have been obtained from the Stock Exchange of Thailand which has compiled the data on total executive compensations and total number of executives from annual registration statements (Form 56-1) submitted by each of its listed companies every year since 2001 (except in 2008 where no data are available) as well as the data from the listed companies’ financial statements. According to the Stock Exchange of Thailand, executives are those who take positions of managing director, deputy managing directors, assistant managing directors, and accounting and finance manager. Compensations include salary, bonus, provident fund, meeting fee, warrant employee joint investment program, warrant employee stock option, as well as other remunerations, for example, car allowance, home rental, and insurance fee. The Stock Exchange of Thailand classifies industry into nine categories. They are agro and food industries, consumer products, finance, industries, property and construction, resources, services, technology, and others. The data from financial statements of the listed companies comprise items such as total assets, total sales, net profits, total revenues, and market capitalization. In our study, we use total assets as a representative for company sizes. By taking natural logarithm of company’s total assets, company size is then classified into ten groups. Due to the availability of the data, our study covers periods between 2002 and 2015. The data on average executive compensation categorized by time period and industry type consist of 5922 observations whereas those grouped by total assets as a proxy for size of company contain 5089 observations. Descriptive statistics for three types of average executive compensation data grouping and for data on companies’ total assets are provided in Table 1. Table 1 Descriptive statistics of average executive compensation and of total assets of companies between 2002 and 2015. The average executive compensation data are categorized based on time period, industry type, and company size. Total assets of companies are used as a representative of company size. Unit: Million Baht Data
Mean
Median
Mode
Maximum
Minimum
Standard deviation
Number of observations
Average executive compensation (time period) Average executive compensation (industry type) Average executive compensation (company size) Total assets of companies
3.880 3.880 3.972 39.425
2.621 2.621 2.662 3.006
0.200 0.200 1.044 6.109
49.198 49.198 49.198 2835.852
0.003 0.003 0.003 0.034
4.342 4.342 4.424 205.657
5922 5922 5089 5089
3. Results Recall that before fitting a representative Lorenz curve to the actual Lorenz plots for three different types of data groupings, namely, time period (T), industry type (I), and company size (C), it is important that we have to test whether or not the distribution of average executive compensation is statistically stable across years so that we can pool the annual data and fit a representative Lorenz curve to the actual Lorenz plots for each type of data groupings. Due to the number of observations for time period (T) and industry type (I) is identical which is equal to 5922, just being categorized in different ways, we can simultaneously test whether or not the actual Lorenz plots that represent the distribution of average executive compensation are statistically different across years for these two types of data categorization. For company size (C), we have to do a separate test since the number of observations is equal to 5089. Tables 2 and 3 report values of D-statistic from the pairwise Kolmogorov–Smirnov test. They indicate that, for each type of data groupings, namely, time period (T), industry type (I), and company size (C), all actual Lorenz plots are not statistically different from each other across years (total of thirteen years) with values of D-statistic between 0.00 and 0.20 and p-values greater than 0.01 in all cases (see Tables S1 and S2 in Supplementary Materials for proportions of companies (x) that fall in the same bins across years (2002–15) grouped by time period (T), industry type (I), and company size (C)). Given these results, we can proceed to pool the data across years and try to fit a representative Lorenz curve to the actual Lorenz plots for each type of data groupings. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Table 2 Values of D-statistic from the pairwise Kolmogorov–Smirnov test show that, for time period (T) and industry type (I), there are no statistical differences in the distribution of average executive compensation across years with p-values, shown in parentheses, greater than 0.01 in all cases.
2002
2002
2003
2004
2005
2006
2007
2009
2010
2011
2012
2013
2014
2015
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.05 (1.000)
0.05 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.05 (1.000)
0.15 (0.965)
0.10 (1.000)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.05 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
2003 2004 2005 2006 2007 2009 2010 2011 2012 2013 2014
0.00 (1.000)
2015
Table 3 Values of D-statistic from the pairwise Kolmogorov–Smirnov test show that, for company size (C), there are no statistical differences in the distribution of average executive compensation across years with p-values, shown in parentheses, greater than 0.01 in all cases.
2002 2003 2004 2005 2006 2007 2009 2010 2011 2012 2013 2014 2015
2002
2003
2004
2005
2006
2007
2009
2010
2011
2012
2013
2014
2015
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.20 (0.771)
0.15 (0.965)
0.20 (0.771)
0.20 (0.771)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.20 (0.771)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.20 (0.771)
0.15 (0.965)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.20 (0.771)
0.10 (1.000)
0.15 (0.965)
0.15 (0.965)
0.00 (1.000)
0.05 (1.000)
0.10 (1.000)
0.15 (0.965)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.05 (1.000)
0.00 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.10 (1.000)
0.00 (1.000)
0.15 (0.965)
0.15 (0.965)
0.10 (1.000)
0.00 (1.000)
0.05 (1.000)
0.05 (1.000)
0.00 (1.000)
0.05 (1.000) 0.00 (1.000)
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Figs. 1–3 show actual Lorenz plots of the distributions of average executive compensation for three different types of data grouping which are time period (T), industry type (I), and size of company (C) between 2002 and 2015 except 2008 (total of thirteen years). For time period (T), there are thirteen Lorenz plots, one for each year (Fig. 1). For industry type (I), the total number of Lorenz plots is one hundred and seventeen (Fig. 2). This number comes from nine industries, each of which has its own Lorenz plot for each year. For company size (C), there are one hundred and twenty-five Lorenz plots (Fig. 3) since we classify company size into ten groups. Each group has its own Lorenz plots for each year. However, there are five groups, each of which comprises only one company. Therefore, we do not count these observations as the Lorenz plot. This would give us one hundred and twenty-five Lorenz plots for company size (C). The actual Lorenz plots illustrated in Figs. 1–3 indicate that the distributions of average executive compensation classified by time period (T) are quite stable during the periods of study relative to those categorized by industry type (I) and company size (C) which show higher variations. Note that a conventional method used to estimate the Lorenz curve could be done by fitting a nonlinear function to the actual observations. However, it is possible that some nonlinear functions may result in the fitted Lorenz curve that is not realistic in a sense that not only does it not pass the coordinate (1, 1) but also gives us a negative value of cumulative normalized average executive compensation (y) for a given value of cumulative normalized rank of companies (x). As described in Section 2.1, there are many nonlinear techniques that could be used to fit the Lorenz curve to the actual observations. However, to keep our method simple and still realistic, we use polynomial function to estimate the Lorenz curve on the conditions that the estimated equation has to be an increasing function and must pass two coordinates which are (0, 0) and (1, 1). We find that the estimated equations for the Lorenz curve for three different types of data grouping could be described by the sixth degree polynomial functions. They are as follows: Time period :
yT = 0.5139x6 + 0.2017x2 + 0.2845x, R2 = 0.9921
(1)
Industry type :
y = 0.4564x + 0.2696x + 0.2740x, R = 0.9770
(2)
Company size :
y = 0.2999x + 0.3423x + 0.3578x, R = 0.9881
(3)
I
C
6
6
2
2
2
2
The estimated Lorenz curves are also illustrated in Figs. 1–3. Note that the estimated Lorenz curve for time period (yT ) fits the actual Lorenz plots reasonably well with R2 = 0.9921 while those for industry type (yI ) and company size (yC ) show higher variances with R2 = 0.9770 and 0.9881, respectively. Based on the three fitted Lorenz curves, we can calculate the Gini coefficient to measure inequality in the distribution of average executive compensation for three types of data grouping. We find that the Gini coefficients for time period, industry type, and company size are equal to 0.4364, 0.4159, and 0.3283, respectively, indicating that inequality of the distribution of average executive compensation is quite equal across different company size relative to across time period and industry type. Our next task is to use the estimated coefficients obtained from Eqs. (1)–(3) above to calculate the proportion of companies on the cumulative normalized rank of companies axis (the x-axis) that corresponds to each bin (total of twenty bins) on the cumulative normalized average executive compensation axis (the y-axis) according to three different types of data grouping. We obtain the proportion of companies that corresponds to each cumulative normalized average executive compensation bin by dividing the cumulative normalized average executive compensation axis (the y-axis) into twenty equal bins (0.00–0.05, 0.06–0.10, 0.11–0.15,. . . , 0.96–1.00). For a given value of cumulative normalized average executive compensation (y), we can use the coefficients from each of the three estimated equations for the Lorenz curve to compute the value of cumulative normalized rank of companies (x), and find out the proportions of companies that fall in each bin (total of twenty bins) on the cumulative normalized rank of companies axis (the x-axis). In order to test the property of scale invariance or self-similarity in the distributions of average executive compensation, we perform the Kolmogorov–Smirnov test to compare whether or not the proportions of companies that fall in the same bin on the cumulative normalized rank of companies axis (the x-axis) according to the three different types of data grouping are statistically different from one another. Our findings are reported in Table 4. The numerical findings from Table 4 indicate that, for each of the cumulative normalized average executive compensation bins, the proportions of companies grouped by time period, industry type, and company size do not differ significantly from each other. Among three types of data groupings, the largest difference is 0.0332 (bin 1) while the smallest difference is 0.0001 (bin 1 and bin 6). The Kolmogorov–Smirnov test results, as shown in Table 4, confirm our numerical findings in that the difference between the distribution of proportions of companies categorized by time period vs. by industry type is statistically insignificant with p-value = 1.000. The difference between the distribution of proportions of companies grouped by time period vs. by company size is also statistically insignificant with p-value = 0.497. In addition, the difference between the distribution of proportions of companies sorted by industry type vs. by company size is statistically insignificant with p-value = 0.771. The results from the Kolmogorov–Smirnov test indicate that there are no statistical differences among the distributions of proportions of companies according to three different types of data grouping for p-values are greater than 0.01 in all cases. Our empirical findings suggest that the distributions of average executive compensation are scale invariance or self-similar. That is time period, type of industry, and company size have no effects on the distributions of average executive compensation. Our results are consistent with what found in previous studies as discussed in Section 1. However, those studies implicate this statistical regularity from fitting power law, gamma, or log-normal distributions to the actual income and wealth observations. We use an alternative method but come up with similar conclusion. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Fig. 1–3. Actual Lorenz plots of the distribution of average executive compensation and estimated Lorenz curves for three different types of data grouping. Cumulative normalized average executive compensation (y) plotted against cumulative normalized rank of companies (x) for three different types of data grouping according to time period (Fig. 1), industry type (Fig. 2), and company size (Fig. 3) between 2002 and 2015. Given the conditions that the fitted Lorenz curve has to be an increasing function and must pass (0, 0) and (1, 1), the estimated equations for the Lorenz curve for each type of data groupings can be described by the sixth degree polynomial functions. Note that the estimated Lorenz curve for time period fits the actual observations reasonably well while those for industry type and company size show higher variances.
Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
T. Sitthiyot, P. Budsaratragoon and K. Holasut / Physica A xxx (xxxx) xxx
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Table 4 Proportions of companies that are in the same bins (total of twenty bins) for three different types of data grouping and results from the KolmogorovSmirnov test. Empirical results show that the distributions of the proportion of companies based on three different types of data grouping are not statistically different from each other with p-values greater than 0.01 in all three cases. Note that the results do not change when the number of bins is ten or forty. Cumulative normalized average executive compensation
0.00–0.05 0.06–0.10 0.11–0.15 0.16–0.20 0.21–0.25 0.26–0.30 0.31–0.35 0.36–0.40 0.41–0.45 0.46–0.50 0.51–0.55 0.56–0.60 0.61–0.65 0.66–0.70 0.71–0.75 0.76–0.80 0.81–0.85 0.86–0.90 0.91–0.95 0.96–1.00
Proportions of companies based on 3 different data groupings
Kolmogorov–Smirnov test results (between 2 groups)
Time period
Industry type
Company size
Time period vs. industry type
Time period vs. company size
Industry type vs. company size
0.1580 0.1326 0.1133 0.0950 0.0778 0.0634 0.0521 0.0436 0.0372 0.0323 0.0285 0.0254 0.0229 0.0209 0.0192 0.0177 0.0165 0.0154 0.0145 0.0136
0.1579 0.1265 0.1066 0.0903 0.0759 0.0635 0.0534 0.0454 0.0391 0.0342 0.0302 0.0271 0.0244 0.0223 0.0205 0.0189 0.0176 0.0164 0.0154 0.0145
0.1248 0.1043 0.0911 0.0811 0.0728 0.0653 0.0585 0.0523 0.0468 0.0419 0.0377 0.0341 0.0309 0.0283 0.0259 0.0239 0.0222 0.0207 0.0193 0.0181
D-statistic = 0.05 p-value = 1.000
D-statistic = 0.25 p-value = 0.497
D-statistic = 0.20 p-value = 0.771
4. Conclusions and remarks While studies about scale invariance or self-similarity in income and wealth distributions have been generally associated with testing whether or not such distributions follow power law, gamma, or log-normal, in this study, we present an alternative method for testing scale invariance or self-similarity. Our method is simple and can be employed to test scale invariance or self-similarity in any types of data, regardless of their distributions. To demonstrate our method, we use the average executive compensation of companies listed in the Stock Exchange of Thailand between 2002 and 2015 as a case study. Despite the complexity of interaction and/or interconnection among companies of different sizes, operating in different industries over a long period of time, each of which at some point may change or adjust its own compensation scheme, as well as a number of major events such as severe flooding, economic recession and recovery, and political instability in Thailand during that period, the empirical evidence on scale invariance in the distribution of average executive compensation found in our study indicates that the overall distributions remain statistically unchanged across time period, industry type, and company size. As argued by West and Brown [9], scale invariance naturally reflects the underlying generic features independently of detailed characteristics of particular systems. Therefore, any theory developed in order to explain the distribution of executive compensation should predict the statistical regularity of scale invariance as previously recommended by Brock [13]. Reference data All data analyzed during this study are included in this article. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Thitithep Sitthiyot: Conceptualization, Methodology, Formal analysis, Writing — original draft, Writing — Review&Editing. Pornanong Budsaratragoon: Investigation, Resources, Writing — Review&Editing. Kanyarat Holasut: Validation, Writing — Review&Editing. Acknowledgments The authors appreciate all reviewers for their useful comments and suggestions. Thitithep Sitthiyot is grateful to Dr. Suradit Holasut for invaluable guidance and fruitful comments. Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.
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Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physa.2019.123556. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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Please cite this article as: T. Sitthiyot, P. Budsaratragoon and K. Holasut, A scaling perspective on the distribution of executive compensation, Physica A (2019) 123556, https://doi.org/10.1016/j.physa.2019.123556.