Zipf's law for share price and company fundamentals
aa r X i v : . [ q -f i n . E C ] F e b Zipf ’s law for share price and companyfundamentals
Taisei Kaizoji and Michiko MiyanoGraduate School of Arts and Sciences,International Christian UnversityFebruary 2, 2017 abstractWe investigate statistically the distribution of share price and the distributionsof three common financial indicators using data from approximately 8,000 com-panies publicly listed worldwide for the period 2004-2013. We find that thedistribution of share price follows Zipf’s law; that is, it can be approximated bya power law distribution with exponent equal to 1. An examination of the dis-tributions of dividends per share, cash flow per share, and book value per share- three financial indicators that can be assumed to influence corporate value(i.e. share price) - shows that these distributions can also be approximated bya power law distribution with power-law exponent equal to 1. We estimate apanel regression model in which share price is the dependent variable and thethree financial indicators are explanatory variables. The two-way fixed effectsmodel that was selected as the best model has quite high power for explainingthe actual data. From these results, we can surmise that the reason why shareprice follows Zipf’s law is that corporate value, i.e. company fundamentals,follows Zipf’s law.
Zipf’s law is an empirical law indicating a relationhip between the frequencyof an event and its rank. It is often formulated by using the complementarycumulative distribution function following a power law,
P r ( X > x ) = cx − α where P r ( X > x ) denotes the probability that a stochastic variable, X , islarger than x , c is a constant and is the power-law exponent. Zipf’s law statesthat the power exponent, α , is equal to unity.In his book, published in 1932 (Zipf 1932), G. K. Zipf reported that thenumber of times that all the various words appear in human language closelyfollows this law : Zipf’s law has since been found to apply in a variety of fields; Zipf’s law for human language (Miller (1958)); Zipf’s law for monkey-typingtexts (Li (1992)), Zipf’s law for the number of people living in a city (Hill 1970,Ijiri and Simon (1977), Gabaix 1999), Zipf’s law for the number of website visits Pareto(1897) before Zipf indicated that the distribution of individual incomes follows apower law distribution.
The data used in this paper come from the OSIRIS database provided by Bu-reau Van Dijk. The database contains financial information on globally listedindustrial companies. We use annual data for the period 2004-2013. After ex-tracting financial data and stock data for 7,796 companies over a 10-year period,we performed a statistical investigation of share price (at closing date) and threefinancial indicators. The financial indicators - dividends per share, cash flow pershare, and book value per share - were obtained by dividing the values providedin the database by the number of shares outstanding.2
Zipf ’slaw for share price
Figure 1 shows the complementary cumulative distribution of share prices plot-ted with logarithmic horizontal and vertical axes. Share prices in this distribu-tion are pooled from the 7,796 companies included in the study for the period2004-2013. In all, there are 47,161 total observations. The right tail of thisdistribution is found to be approximately linear. That is, the distribution canbe approximated by a power law distribution, written as
P r ( X > x ) = cx − α (1)where c is a constant and α denotes the power-law exponent.Figure 1: Zipf’s law for share price: The complementary cumulative dis-tribution of share prices (log-log plot) based on 47,161 share price observationspooled from 7,796 companies for the period 2004-2013. The power-law exponent,estimated using the MLE (maximum likelihood estimator) method, is 1.003.As a first step, we estimated the power-law exponent, Although several esti-mators can be used to estimate the value of the power-law exponent, this paperuses the maximum likelihood estimator (MLE), which is commonly applied.The maximum likelihood estimator of exponent α is ˆ α = n [ n X i =1 ln( x i x min )] − (2)where, x i , i = 1 , · · · , n are the observed values of such that x i ≥ x min ,and ˆ α denotes the estimate of α . Using the share price data pooled from 7,796companies for the period 2004-2013, the estimated power-law exponent is veryclose to unity, ˆ α = 1 .
003 . The results are presented in the first column of Table1. We then test whether that complementary cumulative distribution of shareprices follows a power law distribution by using a goodness-of-fit test basedon a measurement of distance between the empirical distribution function andthe fitted distribution function. The distance is commonly measured either by asupremum norm or a quadratic norm. The most well-known supremum statisticis the Kolmogorov-Smirnov statistic written as D = sup x | F n ( x ) − F ( x ) | (3)On the other hand, the Cramer-von Mises family statistic in a quadratic normis written as Q = n Z ∞−∞ [ F n ( x ) − F ( x )] ψ ( x ) dF ( x ) (4)When ψ ( x ) = 1 , is the Cramer-von Mises statistic, denoted W . Althougha number of other of goodness-of-fit statistics have been proposed, we use thestatistic of Cramer-von Mises in this study . The null hypothesis that theempirical distribution function is a power law distribution is rejected when asmall p -value is obtained. In our study, the null hypothesis is not rejected at the Details of the derivations are given in Appendix Clauset, et. al. (2009) found that Anderson and Darling statistics are highly conservativein their application, reducing the ability to validate the power law model unless there aremany samples in the tail of distribution.
30% significant level since we obtain a p -value of 0.132. That is, the Cramer-vonMises test statistic indicates that the distribution of share prices follows a powerlaw.In the second step, we verify that share price follows Zipf’s law using twotest procedures. One is the Lagrange Multiplier (LMZ) test proposed by Urzua(2000); the other is the Likelihood Ratio (LR) test. The Likelihood Ratio testcan be used in the case of large samples, while the Urzua test can be appliedto cases involving small samples . Test results are shown in Table 1. In bothtests, the null hypothesis that the power-law exponent is equal to 1 cannot berejected at the 10% significance level since the Chi-squared test statistics aresufficiently small.In summary, we can confirm that the distribution of share price follows Zipf’slaw in the right tail, which includes 2% of the total observations.power-law X min Cramer-von Lagrange Likelihood Tailexponent Mises test Multiplier Ratio p -value LMZ LR n1.003 133.4 0.132 0.013 0.010 943Table 1: Test for Zipf’s law for share price: Estimation of the power-law expo-nent of share price, and test results for the power law distribution and for Zipf’slaw. LMZ and LR test involve test-statistics of the Chi-squared distribution.The null hypothesis that the power law exponent is equal to 1 cannot be rejectedat the 10% significance level. In the previous section, it was shown that Zipf’s law holds for share prices. Thissection investigates the relationship between share price and company funda-mentals. To that end, we use the panel regression model proposed by Kaizojiand Miyano (2016b) to estimate company fundamentals. Since the databaseused in this research contains cross-sectional data for the period 2004-2013,panel analysis is appropriate. The model formulates the relationship betweenshare prices and three financial indicators - dividends per share, cash flow pershare, and book value per share - commonly used in models that evaluate cor-porate valuation, so-called company fundamentals. The econometric model canbe formally written asln Y it = a + b ln X ,it + b ln X ,it + b ln X ,it + u it i = 1 , · · · , N ; t = 1 , · · · , T (5)where Y it denotes the dependent variable (share price) for company i in year t ; a denotes a constant; X ,it is the dividends per share of company i in year t ; X ,it is the cash flow per share of company i in year t ; X ,it is the book valueper share of company i in year t ; u it denotes the error term. Urzua(2000) uses the table for significanct points for LMZ. The table is presented in theApendix.
4e estimate the model in equation (5) using the Panel Least Squares method.In the panel regression model, the error term, u it , can be assumed to be dividedinto a pure disturbance term and an error term due to other factors. Assuminga two-way error component model with respect to error, the factors other thandisturbance are (i) factors due to unobservable individual effects, and (ii) factorsdue to unobservable time effects. That is, the error term can be written as u it = µ i + γ t + ǫ it (6)where µ i denotes unobservable individual effects, γ t denotes unobservabletime effects, and ǫ it denotes pure disturbance.If both µ i and γ t are equal to zero, equation (5) is estimated using the pooledOLS method. If either µ i or γ t is equal to zero, equation (6) is a one-way errorcomponent model. If both µ i and γ t are not equal to zero, equation (6) is a two-way error component model. There are two estimation methods for estimatingthe error term in equation (6). One is fixed effects estimation and the otheris random effects estimation. Therefore, the available estimation models are apooled OLS, an individual fixed effects model, a time effects model, a two-wayfixed effects model, an individual random effects model, a time random effectsmodel, and a two-way random effects model .We estimated the models described above and found, after appropriate modelselection tests, that the two-way fixed effects model was the best model. Themodel selection tests used in this study include the likelihood ratio test and F-test for the selection of the pooled OLS model vs the fixed effects model, and theHausman test for the selection of the random effects model vs the fixed effectsmodel. The selection test for the pooled OLS model vs the random effects modelis based on the simple test proposed by Wooldridge (2010) The two-way fixedeffects mode is written asln Y it = a + b ln X ,it + b ln X ,it + b ln X ,it + ǫ it (7) a = a + µ i + γ t where a is a constant term common to all companies, µ i denotes the individualfixed effects, and γ t denotes the time fixed effects. µ i is constant toward timeseries and γ t is constant toward cross section. ǫ it is the pure disturbance. Theindividual fixed effects, µ i , account for an individual company’s heterogeneityand includes such factors as the company’s diversity of corporate governanceand the quality of its employees. The time fixed effects, γ t , indicate variablesthat fluctuate over time but are fixed across companies. The time fixed effectsreflect various shocks, including financial shocks.Table 2 shows the results of the Kaizoji and Miyano (2016b) panel regressionmodel described in equation (7). The signs of the three coefficients are allpositive, which is consistent with corporate value theory. The p -values for thecoefficients are quite small, indicating statistical significance in all three cases.In addition, the R-squared value is 0.97, indicating that the estimated modelexplains much of the variation in share prices. We used the EViews software package to estimate the models. The two-way randomeffects model was unavailable since we used unbalanced panel data. Wooldridge (2010, p.299) proposed the method that uses residuals from pooled OLS tocheck the existence of serial correlation.
50 b1 b2 b3 R Coefficient 1.485 0.137 0.298 0.378 0.969Std. error 0.014 0.003 0.004 0.007 p -value 0.000 0.000 0.000 0.000Table 2: Results of the Panel Regression model (two-way fixed effects model)provided by Kaizoji and Miyano (2016b). Total panel (unbalanced) observationsare 47,161. The p -values for the coefficients indicate statistical significance inall cases. The R-squared value is 0.97.The model (7) is found to have quite high explanatory power with respect toshare price, as the results shown above indicate. From a business managementpoint of view, companies can maximize their share price by enhancing dividendsper share, cash flow per share, and book value per share.Estimates of the two-way fixed effects model for share price, ln ˆ Y it , can beshown as ln ˆ Y it = ˆ a + ˆ µ i + ˆ γ t + ˆ b ln X ,it + ˆ b ln X ,it + ˆ b ln X ,it (8)We call ˆ Y it the theoretical share price. In the previous section, we found that the theoretical share price as estimatedusing a two-way fixed effects model fits actual share price very well, indicatingthat the two-way fixed effects model explains actual share price quite well.However, this result does not explain the reason why the distribution of shareprice follows Zipf’s law. This section examines company fundamentals, definedas the optimal share price reflecting corporate value.Kaizoji and Miyano (2016b) proposed computing company fundamentalsby eliminating the time fixed effects term, , from the theoretical share pricedescribed in equation (8),ln ˜ Y it = ˆ a + ˆ µ i + ˆ b ln X ,it + ˆ b ln X ,it + ˆ b ln X ,it (9)where ˜ Y it denotes the fundamentals of company i in year t .Figure 2 shows the complementary cumulative distributions of actual shareprice and company fundamentals plotted with logarithmic horizontal and ver-tical axes. At a glance, Figure 2 suggests that the distribution of fundamentalsmatches substantially the distribution of actual share prices. Indeed, the two-sample Kolmogorov-Simirnov test leads us to accept the null hypothesis affirm-ing the coincidence of actual share price and company fundamentals (K-statistic= 0.007; p -value = 0.253). In short, the complementary cumulative distributionof share price coincides with that of company fundamentals.We next estimate the power-law exponents using the MLE (maximum like-lihood estimator) method. Comparing the power-law exponent of companyfundamentals with that of actual share price and that of the theoretical shareprice, we found that the three power-law exponents are all close to unity: 1.003for the actual share price, 1.012 for the theoretical share price, 1.006 for thefundamentals. 6e then use the Cramer-von-Mises test to determine whether the comple-mentary cumulative distributions of share prices, theoretical share prices asdescribed in equation (8), and company fundamentals as described in equation(9) are power law distributions. Large p -values in each of the tests indicatethat, for each of the three cases, the null hypothesis affirming that the comple-mentary cumulative distribution is a power law distribution cannot be rejected.(The p -values were 0.132 (actual share price), 0.106 (theoretical share price),and 0.128 (company fundamentals).)Finally, we investigate whether company fundamentals and theoretical shareprice follow Zipf’s law. In both cases, both the Urzua (2000) test and the Like-lihood Ratio test do not reject, at the 10% significance level, the null hypothesisaffirming that the distribution follows Zipf’s law (as was also true for actualshare price). In addition, we found that the complementary cumulative distri-bution of the theoretical share price and that of company fundamentals followZipf’s law in the right tail, which includes 2% of the total observations.Table 3 summarizes the results of estimating power-law exponents and theresults of tests for power law and Zipf’s law for actual share prices, theoreti-cal share prices, and company fundamentals. The data are pooled from 7,796companies for the period 2004-2013; total observations are 47,161.Share price power-law X min Cramer-von- Lagrange Likelihood Tailexponents Mises test Multiplier Ratio p -value LMZ LR nactual 1.003 133.4 0.132 0.013 0.010 943theoretical 1.012 128.5 0.106 0.187 0.136 943fundamental 1.006 119.7 0.128 0.032 0.032 1,000Table 3: Test for Zipf’s law for share price: LMZ and LR are test-statistics ofthe Chi-squared distribution. The null hypothesis affirming that the power lawexponent is equal to 1 cannot be rejected at the 10 % significance level.In summary, we can confirm that the distributions of share price and com-pany fundamentals follow Zipf’s law in the right tail, which includes 2% of totalobservations.From the results described above, it is verified that the complementary cu-mulative distribution of fundamentals obtained from equation (9) closely co-incides with the complementary cumulative distribution of actual share price.Therefore, we can infer that Zipf’s law for share price is caused by Zipf’s lawfor company fundamentals.Figure 2: The complementary cumulative distribution of actual share priceand company fundamentals (log-log plot). Black indicates actual share price, redindicates company fundamentals. The complementary cumulative distributionof fundamentals coincides statistically with that of actual share price. In the previous section, we showed that the distribution of company fundamen-tals, estimated using a two-way fixed effects model, coincides with the distri-7ution of actual share price. This result suggests that company fundamentalssignificantly affect actual share price. However, it does not explain the reasonwhy company fundamentals follows Zipf’s law. To consider the reason why Zipf’slaw for company fundamentals holds, we statistically investigate the distribu-tions of the explanatory variables in the two-way fixed effects model describedin equation (7).Figures 3 through 5 show the plots of the complementary cumulative dis-tributions for the three per share financial indicators using pooled data for theperiod 2004-2013. Figure 3 shows the distribution of dividends per share, Fig-ure 4 shows the distribution of cash flow per share, and Figure 5 shows thedistribution of book value per share. It is obvious that the right tails of thethree distributions can be approximately linearized.Following the same procedure that was used for share price and companyfundamentals, we estimated the power-law exponent for each of the three cases,producing estimates of 1.015 for dividend per share, 1.051 for cash flow per share,and 0.955 for book value per share. For all three distributions, Cramer-von-Mises tests indicate that the null hypothesis affirming that these are power-lawdistributions cannot be rejected ( p -values are, 0.130, 0.151, and 0.221, respec-tively). Moreover, results from applying Urzua’s test (LMZ) and the LikelihoodRatio (LR) test indicate that, in all three cases, the null hypothesis affirmingpower-law exponents cannot be rejected at the 10% significance level. Table 4presents the results. In summary, all three financial indicators per share followZipf’s law.From the results described here, we can infer that the reason Zipf’s lawholds for company fundamentals is that the distributions of the three financialindicators per share, representing corporate value, each follow the power lawdistribution with a power-law exponent equal to 1. Thus, we can conclude thatthe reason Zipf’s law holds for company fundamentals is that Zipf’s law holdsfor dividends per share, cash flow per share, and book value per share.Figure 3: Zipf’s law for dividends per share. The complementary cumulativedistribution of dividends per share (log-log plot) using 47,161 observations basedon pooled data from 7,796 companies for the period 2004-2013. The power-lawexponent estimated by the MLE (maximum likelihood estimator) method is1.015.Figure 4: Zipf’s law for cash flow per share. The complementary cumula-tive distribution of cash flow per share (log-log plot) using 47,161 share priceobservations based on pooled data from 7,796 companies for the period 2004-2013. The power-law exponent estimated by the MLE (maximum likelihoodestimator) method is 1.051.Figure 5: Zipf’s law for book value per share. The complementary cumula-tive distribution of the book value per share (log-log plot) using 47,161 shareprice observations based on pooled data from 7,796 companies for the period2004-2013. The power-law exponent estimated by the MLE (maximum likeli-hood estimator) method is 0.955. 8nancial indicator power-law X min Cramer-von- Lagrange Likelihood Tailper share exponents Mises test Multiplier Ratio p -value LMZ LR nDividends 1.015 3.6 0.130 0.946 0.292 1250Cash Flow 1.051 21.9 0.151 3.809 2.322 943Book Value 0.955 98.9 0.221 2.828 2.010 943Table 4: Tests for Zipf’s law for the three financial indicators. LMZ and LRare test-statistics of the Chi-squared distribution. The null hypothesis affirm-ing that the power-law exponent is equal to 1 cannot be rejected at the 10%significance level. In Section 3, we showed that our measure of company fundamentals followsZipf’s law. This means that there exists an extreme disparity in company fun-damentals. In this section, we propose a model to explain the extreme disparityin company fundamentals from a theoretical point of view. It is easy to imaginethat there are various factors explaining company fundamentals. For example,the quality of company employees, the foresight of company executives, produc-tion technology, and corporate governance can be listed as contributing factors.The formation of corporate value has its origin in the compound multiplier ef-fects of these factors. Thus, the disparity in company fundamentals can beconsidered a reflection of differences in those factors, their number and influ-ence, that affect corporate value. It can be assumed that there are a relativelyfew companies that have an abundance of strongly positive factors, while a greatmany companies have few such positive factors.We propose a simple model which formalizes this notion . Consider avariable, Z n , which describes a factor that affects the fundamentals of a com-pany.If Z n is greater than 1, the factor represented will enhance company funda-mentals. Inversely, if Z n < Assumption 1 (Compound multiplier effects): Z n represents a setof stochastic variables that follow identical independent distributions. We canassume that company fundamentals can be determined by multiplying the Z n variables. That is, the value of company fundamentals is defined by the equation X N = X ( Z Z Z · · · Z N − ) = X N − Y n =0 Z n (10)where X denotes the value of company fundamentals at the initial point. Assumption 2:
The number, N , of stochastic variables, Z n , differs bycompany. Furthermore, we assume that N follows a geometric distribution.That is, f N ( n ) = P r ( N = n ) = pq n − f or n = 1 , , · · · (11) The model proposed here is mathematically the same as the model of income distributionsproposed by Reed (2004) < p < q = 1 − p . Thus, the value of company fundamentals, ¯ X = X Q N − n =0 Z n , is a variable, where N is a random variable following (10) .Given Assumption 1 and Assumption 2, we have the following Proposition:(i) If the probability that variable Z n is greater than one is non-negative,that is, P r ( Z n > > X n can be approximated by the following power law distribution: P r ( ¯
X > ∼ cx − α , as x → ∞ (12)where c and α are positive constants.(ii) If λ = pq →
0, then α → This paper considers the reason why share prices follow Zipf’s law. To thisend, we investigate the relationship between company fundamentals and shareprice. We use a database containing financial information for approximately8,000 globally listed companies and estimate company fundamentals using apanel regression model (two-way fixed effects model).We find that the distribution of company fundamentals follows Zipf’s law,and, moreover, that the distribution of fundamentals matches substantially thedistribution of actual share prices. From these findings, we conclude that Zipf’slaw for share prices reflects Zipf’s law for company fundamentals. More gen-erally, to the extent that the stock market has the ability to properly evaluatecompany fundamentals, the distribution of share price reflects company fun-damentals. We also find that three common financial indicators - dividendsper share, cash flow per share, and book value per share - follow Zipf’s law.These findings appear to suggest that Zipf’s law for company fundamentals hasa robust disposition.We show a simple stochastic model to explain Zipf’s law for company fun-damentals. However, the question of why the extreme intercompany disparitiesdescribed by Zipf’s law exist in a capitalistic economy remains mysterious andunexplored.
Appendix
A: Table of significance values for LMZ for use in Urzua(2000)
Source: Own Monte Carlo simulation using inversion method, and after 1000,000replications 10 10 15 20 25 30 50 100 200 ∞ Level5% 6.19 6.14 6.09 6.08 6.03 5.98 5.98 5.99 5.9910% 4.38 4.41 4.43 4.45 4.46 4.49 4.56 4.58 4.61
B: Derivations of the maximum likelihood estimator (MLE)for the shape parameter of a power law.
In equation (1) , we described a power law distribution as a complementarycumulative distribution with power-law form. The probability density functionfor the Pareto distribution is defined as f ( x ) = αk α x α +1 x ≥ k > α is the shape parameter and k is the scale parameter correspondingto the minimum value of the distribution.The probability density function, f ( x ) , is given by the following likelihoodfunction L = n Y i =1 αk α x α +1 i (B.2)Logarithm L of the likelihood function is written asln L = ln n Y i =1 αk α x α +1 i = n X i =1 [ln α + α ln k − ( α + 1) ln x i ]= n ln α + nα ln k − ( α + 1) n X i =1 ln x i (B.3)Setting ∂L/∂α = 0 and solving for α , we obtain the following MLE for theshape parameter. nα + n ln k − n X i =1 ln x i = 0 (B.4) α = n [ n X i =1 ln( x i k )] − (B.5)Let ˆ k = min i x i , then ˆ α = n [ n X i =1 ln( x i x min )] − (B.6) The Pareto distribution is equivalent to a cumulative distribution with power law form : Sketch of a Reed’s (2004) proof of the Proposition We derive the Paretian tail behavior for the model. The derivation uses gener-ating functions the probability generating function (pgf), which for a discreterandom variable X with pmf, is defined as G X ( s ) = E ( s X ) = X f x ( x ) s x (C.1)and the moment generating function (mgf), which for any random variable X isdefined as M X ( s ) = E ( e sX ) (C.2)provided the expectations exist. For a random variable, N, with a geometricdistribution (10), the pgf is G N ( s ) = ps − qs (C.3)Now let ¯ Y = log( ¯ X ) , where ¯ X is a random variable denoting the funda-mentals: ( ¯ X = X Q N − i =0 Z i ) . Then¯ Y = Y + N − X i =0 U i (C.4)where Y = log( X ) and U i = log( Z i ) for i = 0 , , · · · , N − Y is M ¯ Y ( s ) = E ( e s ¯ Y ) = E (exp[ Y s + N − X i =0 U i s ]) (C.5)where the expectation is taken with respect to the random variables Y , N and U , U , · · · , U N − , assumed to be independent. Using conditional expecta-tion, this can be written as M ¯ Y = M ( s ) E (exp[ Y s + N − X i =0 U i s ]) (C.6)where M ( s ) is the mgf of Y ; M U ( s ) is the common mgf of the U i and the ex-pectation is taken with respect to the random variable N . Now, E ([ M U ( s )] N ) = G N ( M U ( s )), so that the mgf of ¯ X can be written M ¯ Y = M ( s ) pM U ( s )1 − qM U ( s ) (C.7)From standard results, the tail behavior of the pdf of ¯ Y can be determinedfrom the singularities of its mgf. These occur at the solution (in s ) to M U ( s ) =1 /q . We examine these in the case where fundamentals are able to increaseand decrease: P r ( Z i > > P r ( Z i < > P r ( U i > > P r ( U i < > M U ( s ) → ∞ as s → ∞ and s → −∞ . From this fact and the convexityof M U ( s ), it follows that there are two simple poles of M ¯ Y ( s ), one positive α and the other negative − β . This implies f ¯ X ( s ) ∼ c x − α − ( as x → ∞ ) and f ¯ X ( s ) ∼ c x β − ( as x →
0) (C.8)12he asymptotic behavior of the cdf F ¯ X ( x ) or the complementary cdf, 1 − F ¯ X ( x ) , follows from integration.We introduce the quantity λ = p/q and the quantity θ = 1 − log( q ) which arerelated as θ = − log(1 + λ ) , an increasing function. It follows that β increaseswith λ . In the limit as λ → , β → − F ¯ X ( x ), tends to follow the power law distribution with the exponent 1 + β equal tounity in the right tail. References − − + Share price x C u m u l a t i v e P r obab ili t y Figure 1: Zipf’s law for share price: The complementary cumulative distributionof the share prices. (log-log plot). The 47,161 share price data is pooled from7,796 companies for the period 2004-2013. The power-law exponent estimatedby MLE (maximum likelihood estimator) method is 1.003.15 e−01 1e+01 1e+03 1e+05 − − + Share price and Fundamental share price x C u m u l a t i v e P r obab ili t y − − + Figure 2: The complementary cumulative distribution of actual share price andfundamental (log-log plot). black: actual share price and red: fundamentals Thecomplementary cumulative distribution of fundamentals coincides statisticallywith that of actual share price − − + Dividends per share x C u m u l a t i v e p r obab ili t yy
0) (C.8)12he asymptotic behavior of the cdf F ¯ X ( x ) or the complementary cdf, 1 − F ¯ X ( x ) , follows from integration.We introduce the quantity λ = p/q and the quantity θ = 1 − log( q ) which arerelated as θ = − log(1 + λ ) , an increasing function. It follows that β increaseswith λ . In the limit as λ → , β → − F ¯ X ( x ), tends to follow the power law distribution with the exponent 1 + β equal tounity in the right tail. References − − + Share price x C u m u l a t i v e P r obab ili t y Figure 1: Zipf’s law for share price: The complementary cumulative distributionof the share prices. (log-log plot). The 47,161 share price data is pooled from7,796 companies for the period 2004-2013. The power-law exponent estimatedby MLE (maximum likelihood estimator) method is 1.003.15 e−01 1e+01 1e+03 1e+05 − − + Share price and Fundamental share price x C u m u l a t i v e P r obab ili t y − − + Figure 2: The complementary cumulative distribution of actual share price andfundamental (log-log plot). black: actual share price and red: fundamentals Thecomplementary cumulative distribution of fundamentals coincides statisticallywith that of actual share price − − + Dividends per share x C u m u l a t i v e p r obab ili t yy Figure 3: Zipf’s law for the dividends per share. The complementary cumulativedistribution of the dividends per share (log-log plot). The 47,161 dividends pershare is pooled from 7,796 companies for the period 2004-2013. The power-lawexponent estimated by MLE method is 1.01516 e−03 1e−01 1e+01 1e+03 − − + Cash Flow per share x C u m u l a t i v e p r obab ili t y Figure 4: Zipf’s law for cash flow per share. The complementary cumulativedistribution of the cash flow per share (log-log plot). The 47,161 cash flow pershare is pooled from 7,796 companies for the period 2004-2013. The power-lawexponent estimated by MLE method is 1.051 − − + Book Value per share x C u m u l a t i v e p r obab ili t yy