On the normality of stock return distributions: Latin American markets, 2000-2007.
The assumption of a normal distribution for the behavior of stock market returns is commonplace in the theory of finance, financial models, and security valuation methodologies. A normal distribution is a two-parameter distribution that consists of the expected value and its variance; thus, the distribution is completely determined by its first two moments. The first moment, the expected value of the return, relates to the return of the security, and the square root of the second central moment, the standard deviation, relates to the risk for the security.
The two moments are evident in the mean-variance model of Markowitz (1976) and the valuation of options described by Black and Scholes (1973). The literature survey on option pricing by Broadie and Detemple (2004) contains further reference to the two moments of a normal distribution. However, after the research efforts of Mandelbrot (1963) and Fama (1965), a mounting quantity of empirical evidence exists against the normality assumption.
If the normality assumption is wrong, many of the models and methodologies applied in the valuation of securities could misestimate the appropriate price of those securities. Misestimating is especially relevant if the appropriate distributions involve three or more parameters. In such cases, the association of the standard deviation with a measure of risk could be misleading.
In terms of the normality assumption, the association of risk with the standard deviation of the distribution implies that consideration of risk is related to return deviations from the mean. For a nonnormal fat-tail distribution, however, risk could relate to the possibility of extreme events. Investors may be perfectly comfortable with return behavior that fluctuates relatively close to its mean value and only become worried when extreme negative values occur. If the normality assumption is incorrect, considering only the standard deviation as a measure of risk is not appropriate.
Given the importance of a better knowledge of the distribution of stock market returns, many researchers have directed their attention to exploring the normal distribution assumption with respect to stock markets after the turn of the century (Aggarwal, Inclan, & Leal, 2001; Ane & Labidi, 2001, 2004; Aparicio & Estrada, 2001; Arbelaez, Urrutia, & Abbas, 2001; Balaban, Ouenniche, & Politou, 2005; Blenman, Chatterjee, & Ayadi, 2005; Chen, Gupta, & Troskie, 2003; Harris & Kucukozmen, 2001; Malevergne, Pisarenko, & Sornette, 2005; Ortiz & Arjona, 2001; Tolikas & Brown, 2006; Wilkens, 2005).
The central limit theorem is the basis of the normality assumption for stock behavior. Many transactions occur on a given day for a specific stock. If price variations between transactions belong to independent identical distributions, a basis exists for the hypothesis of a normal distribution for the daily returns in terms of the central limit theorem. However, to apply the central limit theorem, one must consider additional elements.
The assumption of normality holds that information arrives uniformly at the market and that the investors react uniformly to the arrival of the information (Aparicio & Estrada, 2001). Whether the two conditions are true is questionable. For a normal distribution, the independent identical distribution generating the price changes in the stock must be of finite variance. Where the latter is not true, a normal distribution cannot be established or assumed.
This article involved testing the normality assumption for daily stock market returns of six Latin American stock markets for the period 2000 to 2007. The research included applying several normality tests. For all of the tests, the normality assumption was rejected for all of the markets considered at demanding levels of statistical significance.
Section 2 includes a presentation of the data used and the preliminary findings against which the normality assumption was tested. Section 3 contains the results of applying the normality tests of Shapiro-Wilk, Jarque-Bera, Pearson chi-square, Shapiro-Francia, and Kolmogorov-Smirnov to the empirical data and an analysis of the findings. A discussion of the discrepancies between the empirical data and adjusted normal distributions appears in Section 4. Section 5 includes a report of the findings for market returns for longer periods, specifically time blocks of 5, 20, 60, and 120 consecutive market days. Summarized in Section 6 are the main conclusions of the article. Appendix A and B graphically illustrate the data analyzed.
Measuring daily market returns involves considering the daily variations of the stock market indices. Specifically, the following market indices were considered: Indice del Mercado de Valores de Argentina (IMERVAL), Argentina; Indice de la Bolsa de Valores de Sao Paulo (IBOVESPA), Brazil; Indice General de Precios de Acciones (IGPA), Chile; Indice General de la Bolsa Colombiana (IGBC), Colombia; Indice de Precios y Cotizaciones (IPC), Mexico; Indice General de la Bolsa de Valores de Lima (IGBVL), Peru; and Indice Bursatil de Caracas (IBC), Venezuela. These markets will hereafter be referred to by their respective countries: IMERVAL is Argentina (ARG), IBOVESPA is Brazil (BRA), IGPA is Chile (CHI), IGBC is Colombia (COL), IPC is Mexico (MEX), IGBVL is Peru (PER), and IBC is Venezuela (VEN).
Table 1 presents a summary of macroeconomic data for the countries considered, including the gross domestic product (GDP) in 2007, rate of growth of the GDP, and the inflation rate in the period under study. The data show the size and performance of the economies for the period 2000 to 2007. In addition, the table indicates the total amount traded in stocks in 2007 for each market and the yearly average of traded amount of stocks for the period considered.
All the countries considered in the study are emerging economies; nevertheless, as evident in Table 1, the countries showed great diversity in the size of their economies. With respect to the GDP of 2007, the smallest economy was Peru with a GDP of $102 billion, and the biggest economy was Brazil with a GDP of $1,295 billion, a ratio of 1 to 12.7. For the period under study, all the countries experienced growth, with average yearly rates varying from 2.9% for Mexico to 4.8% for Peru. Except for Venezuela, all the countries experienced relatively low levels of inflation. The last two columns show the size of the countries' stock markets, all of which are small markets. The biggest market was Brazil with a total traded amount of $608 billion, and the smallest market was Venezuela with $1.1 billion of total transactions in 2007.
The time-series returns were defined as [R.sub.t] = (ln([I.sub.t]) - ln([I.sub.t]-1)), where [R.sub.t] is the return on day t; ln([I.sub.t]) is the natural logarithm of [I.sub.t]; and [I.sub.t] is the stock market index at the end of day t. Thus, [R.sub.t] corresponds to a continuously compounded rate. The sample period consisted of all the daily returns generated by the stock markets from January 1, 2000, to December 31, 2007. Each country's stock market website provided the data.
Appendix A (see Figures A1-A14) illustrates the behavior of each stock market index and its respective return. The behavior over time of the returns is given in terms of standardized returns, in other words, in terms of units of standard deviations with respect to the mean. The figures show some variations in the size of the return fluctuations, especially for Argentina. The variations signal possible heteroscedasticity. In the case of the statistics for Argentina, the heteroscedasticity is clear: High variations in the period from the last semester of 2001 to the first semester of 2002 were apparent and may be due to the economic crisis experienced by Argentina.
Table 2 provides a summary of the empirical distributions of the stock market returns considered. The statistics include the mean (M), the standard deviation (SD), the minimum and the maximum values, and the coefficients of skewness and kurtosis. With the exception of Argentina, all markets exhibited negative coefficients of skewness, signaling the possibility of left asymmetric distributions and a mean value lower than the mode and the median. The lowest coefficient of skewness corresponded to the stock market of Venezuela. Negative skewness means a higher probability of extreme negative returns relative to extreme positive returns, which is in contrast to the prediction of symmetric distributions such as the normal distribution that shows equal probabilities for extreme cases. The asymmetrical behavior evident for returns implies serious shortcomings when considering the standard deviation as the sole measure of risk.
Table 2 shows that for all the markets, values for kurtosis were well above zero, pointing toward a nonnormal distribution. The lowest kurtosis corresponded to Brazil and the highest to Venezuela. The values for kurtosis relate to leptokurtic distributions, which exhibit fat tails and high peaks with respect to normal distributions. When considering a possible nonnormal distribution characterized by negative asymmetry and fat tails, a situation of higher probabilities for extreme negative returns emerges than would in the case of a normal distribution. Leptokurtic distributions severely restrict the appropriateness of using the standard deviation of a distribution as the sole means of identification of risk.
For a normal distribution, the coefficients of skewness and kurtosis are asymptotically normally distributed with an expected value of zero and a variance of 6/N and 24/N respectively, where N is the sample size. Values for the standardized coefficients outside the range of -1.96 and 1.96 would signal a departure from a normal distribution at the p < .05 significance level. Table 3 indicates the standardized values for skewness and kurtosis.
All the markets but Colombia reflected levels of skewness significantly different from zero, and only Argentina showed a positive skewness. With respect to kurtosis, all the markets demonstrated leptokurtic distributions. The histograms in Appendix B (see Figures B1-B7) illustrate the results along with the normal standardized distributions corresponding to the samples' means and variances. The following section involves a discussion of the rejection of the normality assumption using the five statistical tests mentioned previously.
The following tests of normality were applied to the time-series data: Shapiro-Wilk, Jarque-Bera, Pearson chi-square, Shapiro-Francia, and Kolmogorov-Smirnov. Table 4 shows the results.
Evident in Table 4, the hypothesis of normality was rejected at the p < .05 significance level for all the markets for all the tests performed. Furthermore, with the exception of Brazil, the rejection of the normality assumption was apparent at a p < .01 significance level. For all the tests conducted, p values were zero up to the fourth decimal place for all the markets except Brazil. Brazil showed a p value of 0.0410 for the Pearson chi-square test; thus, the null hypothesis of normality could be rejected at a p < .05 significance level but not at a p < .01 significance level.
Discrepancies between Empirical Data and Adjusted Normal Distributions
The previous section indicated significant departures from the norm expected for stock returns. This section involves estimating the size of the errors when assuming a normal distribution for the returns of the stock markets under study. For this purpose, for each market, the empirical distribution was compared with a normal distribution, with both distributions obtained from the empirical data. The normal distributions were generated by adjusting the first two central moments to the sample estimates in accordance with the empirical data.
Table 5 shows the probability discrepancies between an adjusted normal distribution of returns and the empirical distribution for different return intervals. Definition of the return intervals involved considering the length of one unit standard deviation. The first column in the table presents the return intervals considered, the second column shows the theoretical probability of occurrence for a normal distribution, and the remaining columns illustrate the empirical probability of the occurrence based on the relative frequency in the empirical data.
Comments will be restricted to intervals corresponding to returns lower than the sample mean in the first half of Table 5. Barring aspects related to features of asymmetry, the same comments could apply to intervals of returns higher than the sample mean. The comparison between the theoretical and empirical probabilities clearly showed a leptokurtic empirical distribution.
Thus, the theoretical probability corresponding to the interval between the sample mean and 1 standard deviation below was lower than the corresponding empirical probabilities for each country. Empirical distributions suggest higher frequencies of occurrence of events near the sample mean than does normal distribution, signaling a peaked distribution relative to the Gaussian curve. In the adjacent interval, or returns 1 and 2 standard deviations below the sample mean, the inverse was apparent: A higher normal theoretical probability relative to each empirical probability for each country was apparent. The following adjacent interval, or returns 2 to 3 standard deviations below the sample mean, illustrated a normal theoretical probability higher than the empirical probabilities for Argentina, Chile, Colombia, Peru, and Venezuela and lower than the empirical probabilities for Brazil and Mexico. For the next adjacent interval, or returns falling 3 and 4 standard deviations below the sample mean, except for Venezuela, all countries reflected higher empirical probabilities than those predicted by a normal distribution. In the two extreme intervals, shown in the table, which corresponded to returns between 4 and 5 and between 5 and 6 standard deviations below the sample mean, all the countries showed much higher empirical probabilities than the prediction of the normal distribution.
Empirical distributions presented fatter left tails than did the normal distribution. Bearing in mind that similar comments can apply to returns to the right of the sample mean, it can be appreciated that empirical distributions for all countries considered were peaked distributions with fatter tails relative to the Gaussian distribution. Thus, leptokurtic distributions seem more appropriate to describe the data for each country.
Comparing the number of outliers predicted for a normal distribution and the actual number of occurrences in the empirical data of each country allowed for a better appreciation of the situation. Table 6 illustrates the number of extreme returns in the empirical data in comparison to the expectations of the corresponding theoretical normal distribution. Extreme values of occurrence were calculated for both positive and negative returns. The first half of the table indicates the situation for negative extreme values and the second half for positive extreme values.
Except for asymmetry features, the behavior of both tails of the distribution was similar. Discussion will be restricted to the left tail. As evident in Table 6, for returns 3, 4, and 5 standard deviations below the mean, the number of empirical data returns was much higher than the number expected for a normal distribution. For instance, Peru reflected 20 cases of returns 3 standard deviations below the mean, whereas the expectation of the normal distribution was 2.6932 cases, a difference of 643%.
For returns 6 standard deviations below the mean, 3 countries showed no occurrences of returns. Actual occurrences can only be in integers, in contrast to theoretically expected cases. Thus, given the small size of the sample and the low probability of occurrences of returns, for the interval considered, determining whether the expected number of cases of the empirical distribution is lower or higher than the number expected in a normal distribution was not possible. However, the pattern exhibited suggests similar behavior for the interval when compared to the former extreme intervals.
Therefore, investors who make investment decisions based on a normal distribution could be underestimating severely the probability of extreme adverse returns. Were investors' risk aversion more related to the possibility of extreme negative returns than frequent moderate negative returns, the assumption of a normal distribution could lead to bad investment decisions for the return-risk preferences of investors.
Normality over Time
This section involves examining whether longer return periods approach a normal distribution. The study involved considering periods of 5, 20, 60, and 120 block days. Calculation of returns occurred as follows: R[K.sub.t] = Ln([I.sub.t]) - Ln([I.sub.t-k]). In the equation, R[K.sub.t] is the market return for period t, corresponding to a period of K block days, with K = 1, 5, 20, 60, and 120 block days. Ln([I.sub.t]) is the natural logarithm of a corresponding market index for day t, and t = K+1, 2K+1, 3K+1, and so on, with k = K, K+1, K+2, and so forth.
The tables that follow indicate the results for the possible conformity to a normal distribution for market returns as block days increase, with two tables appearing for each country. The first table for each country provides estimations of sample moments of the distributions for K = 1, 5, 20, 60, and 120. More specifically, the tables illustrate data related to sample size, mean return, standard deviation, minimum and maximum returns, skewness, standardized skewness, kurtosis, and standardized kurtosis. Section 2 included definitions of the last four statistics. The second table for each country shows the results of the normality tests applied to the different time series of returns. More specifically, the tables present the statistics and the p values corresponding to the following normality tests: Shapiro-Wilk, Jarque-Bera, Pearson chi-square, Shapiro-Francia, and Kolmogorov-Smirnov.
The first tables for each country allow for the analysis of symmetry and kurtosis for the empirical distributions relative to a normal distribution. Asymptotically, a normal distribution implies that the coefficients of skewness and kurtosis will follow a normal distribution with expected values of zero and variances of 6/N and 24/N respectively, where N is the sample size. Thus, if the coefficients of standardized skewness and standardized kurtosis are outside of the interval of -1.96 and 1.96, one may question the normality assumption. The second tables for each country illustrate formal tests with respect to the normality assumption. Conclusions were drawn based on a p < .05 significance level.
With respect to skewness, Table 7 shows departures from a normal distribution for 1, 20, and 120 market days. Similar conclusions cannot be reached for 5 and 60 market days. In addition, the coefficients of standardized skewness exhibited positive and negative signs for different time blocks. Thus, for Argentina, no assertion can be made with respect to a convergence toward a symmetrical empirical distribution as block days increase.
In Table 7, a decreasing standardized kurtosis as the block of time increases is noticeable. The possibility of a kurtosis behavior corresponding to a normal distribution could not be rejected for periods longer than 60 market days. Thus, from the information in Table 7, one can conclude that for Argentina, the longer the time block, the less peaked and less fat tailed the market return distribution. No conclusions could be reached with respect to symmetry.
Table 8 shows that, with the exception of the Pearson chi-square test, all the other tests rejected the normality assumption for periods of 20 or fewer market days. For 60 market days, the normality assumption could not be rejected based on any of the tests. For K = 120, the Shapiro-Wilk test and the Shapiro-Francia test indicated rejection of the normality assumption, whereas the Jarque-Bera, the Pearson chi-square, and the Kolmogorov-Smirnov tests did not permit rejection of the normality assumption. In general, the empirical data indicate a nonnormal distribution for the short time horizon and a behavior that could be approaching a normal distribution as the time horizon increases.
Table 9 illustrates coefficients of standardized skewness. The coefficients indicated the symmetrical behavior of the distribution for all time blocks but that daily returns could not be rejected. With respect to kurtosis, the hypothesis of normality could not be rejected for time blocks of 20 or more market days. The result suggests the possibility of market return distribution approaching a normal distribution as the period increases.
As evident in Table 10, all the tests rejected the normality assumption for K = 1. For K = 5, the tests of Jarque-Bera and Shapiro-Francia led to rejection of the normality assumption. In contrast, the Shapiro-Wilk, the Pearson chi-square, and the Kolmogorov-Smirnov tests did not permit rejection of the normality assumption. For K = 20, 60, and 120, no tests allowed for the rejection of the normality hypothesis, except for the Pearson chi-square test for 120 block days.
In accordance with the data in Table 11, symmetrical behavior could not be rejected for time blocks of 20 or more market days. Levels of kurtosis of zero could not be rejected for time blocks greater than 60 market days. The results indicate the possibility of market return behavior approaching a normal distribution as the length of time increases.
Table 12 shows that the normality assumption was rejected for K = 1 and K = 5 for all tests performed. For K = 20, only the Jarque-Bera test permitted rejection of the normality assumption; all the other tests did not reject the normality assumption. For K = 60 and K = 120, no test performed allowed for the rejection of the normality assumption.
For periods of 60 market days and longer, Table 13 illustrates levels of coefficients of skewness and kurtosis that could not lead to rejecting the hypothesis of a normal distribution. A possibility exists that as the length of time increases, the distribution of market returns approaches a normal distribution.
As evident in Table 14, all the tests permitted rejection of the normality assumption for K = 1 and K = 5. For K = 20, only the Jarque-Bera test allowed for rejection of the normality assumption; none of the other tests rejected the normality assumption. For K = 60 and K = 120, no test led to rejection of the normality assumption.
As apparent in Table 15, for periods of 120 market days, the coefficient of skewness could not lead to a rejection of the normality hypothesis. With respect to kurtosis, for K = 60 and K = 120, a normal distribution cannot be excluded.
Table 16 shows that the normality assumption was rejected under all the tests performed for K = 1, K = 5, and K = 20. For K = 60, except for the test of Jarque-Bera, all tests permitted rejection of the normality assumption. For K = 120, no test allowed for the rejection of the normality assumption.
Table 17 illustrates that for periods of 60 and 120 market days, coefficients of skewness and kurtosis could not lead to the rejection of the possibility of a normal distribution.
As evident in Table 18, all tests permitted rejection of the assumption of normality for K = 1 and K = 5. For K = 20, the tests of Shapiro-Wilk, Jarque-Bera, and Shapiro-Francia allowed for the rejection of the normality assumption, whereas the other tests did not. For K = 60 and K = 120, no test led to the rejection of the normality assumption.
For periods of 60 and 120 market days, Table 19 illustrates coefficients of skewness and kurtosis that did not support rejection of the possibility of a normal distribution.
Table 20 indicates that for K = 1, K = 5, and K = 20, all tests permitted rejection of the normality assumption. For K = 60 and K = 120, none of the tests allowed for rejection of the normality assumption.
In general, for shorter periods, the empirical distributions of all the countries departed from a normal distribution and reflected asymmetric and leptokurtic behavior, a left-tailed distribution peaking around the mean value with fat tails. For intermediate values of K for the countries considered, some tests permitted rejection of the normality assumption, and others did not. For lengthier blocks of time, none of the tests supported rejection of the assumption of normality for any country; neither the hypothesis of symmetry nor null kurtosis was rejected. Thus, for the countries considered, returns distributions are nonnormal over short periods and approach a normal distribution when measured over longer periods.
For the Latin American stock markets of Argentina, Brazil, Chile, Colombia, Mexico, Peru, and Venezuela for the period 2000 to 2007, the daily market returns showed important departures from a normal distribution. The empirical data indicated that leptokurtic probability distributions are a better explanation of the behavior of daily market returns for the stock markets over the period under study. Specifically, the best description of the empirical distributions of daily market returns for all the markets would be peaked distributions around the mean with fat tails, which, except for Argentina, reflected an asymmetry that leaned toward the left.
The data for all the markets in the study led to a rejection of the null hypothesis of normal daily market returns for all the tests performed (Shapiro-Wilk, Jarque-Bera, Pearson chi-square, Shapiro-Francia, and Kolmogorov-Smirnov) at the p < .05 significance level. Furthermore, with the exception of Brazil, the assumption of normality was rejected at the p < .01 significance level.
The findings indicate that the assumption of normal daily market returns could lead to significant underestimations of the probability of extreme returns occurring. If investors' risk aversion were focused particularly on extreme negative returns, decisions based on the assumption of a normal distribution of returns would not produce suitable investment returns for the investors' risk preferences. Moreover, fat-tail distributions could imply serious limitations for using the standard deviation of the distribution to predict the risk associated with returns. From the point of view of asset valuation, the assumption of normality for market returns could produce misestimates of the risk-return relationship and affect the composition of efficient portfolios.
When considering longer periods for the market returns, the empirical distributions reflected a behavior closer to a normal distribution. Thus, the longer the block of days measured for market returns, the closer the empirical distributions of the countries under study approximated a symmetrical and mesokurtic distribution. In general, the conclusion applies to all the markets in the study, although the results present some peculiarities depending on the country considered and the particular test. Results showed differences related to the length of the time for which the assumption of normal distribution could not be rejected. At the p < .05 significance level for all the tests performed, except the Pearson chi-square test, the normality hypothesis could not be rejected from 20 market days for Brazil; 60 for Chile, Colombia, Peru, and Venezuela; and 120 for Mexico.
Argentina is a special case. The tests supported rejecting the normality hypothesis for 1, 5, and 20 market days, but for 60 market days, no test indicated rejection of the hypothesis. For the 120 market day block, the Shapiro-Wilk test and the Shapiro-Francia test permitted rejection of the Gaussian hypothesis, but the other tests could not support rejection of the assumption of normality. Whether the different behavior of Argentina is due to the financial crisis experienced by the country between 2001 and 2002 remains an open question beyond the scope of this article.
Further research is required at both the theoretical and empirical levels. At the theoretical level, the reasons behind the nonnormal behavior of market returns distributions for short periods, despite the central limit theorem, require exploration and explanation. At the empirical level, testing theoretical distributions against empirical data, especially in the case of emerging capital markets, is important.
Appendix A: Market Behavior
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Appendix B: Histograms
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Sergio J. Chion Professor, CENTRUM Catolica, Pontificia Universidad Catolica del Peru Doctor of Philosophy, Stanford University, USA Master of Science, Stanford University, USA
Carlos N. Veliz C. ** Professor, CENTRUM Catolica, Pontificia Universidad Catolica del Peru Doctor en Ciencias, Pontificia Universidad Catolica del Peru Doctor en IngenieriaIndustrial, Universidad Nacional Mayor de San Marcos, Peru Magister en Matematicas, Pontificia Universidad Catolica del Peru
* Special thanks go to Esteban Hnyilicza for his comments and suggestions. Thanks also go to the anonymous referees of the journal. We appreciate the general interest of Mario Tello during the development of this article and thank Ross Mary Rodriguez for research assistance. Any errors that may occur are solely ours.
** Correspondence with the authors to firstname.lastname@example.org and email@example.com
Table 1 Figures for the Period 2000-2007 Real growth of GDP Inflation (b) GDP (a) 2007 2000-07 (yearly 2000- 07 (yearly Market (billion $) average rate) average rate) ARG 248 3.1% 9.2% BRA 1,295 3.3% 7.0% CHI 161 4.5% 3.4% COL 172 4.1% 6.3% MEX 886 2.9% 4.9% PER 102 4.8% 2.2% VEN 227 4.3% 19.5% Total traded amount Market of stocks in 2007 Yearly average of traded amount (million $) of stocks (c) 2000-07 (million $) ARG 7,372 5,874 BRA 608,432 174,931 CHI 49,900 16,939 COL 16,857 5,446 MEX 123,914 59,641 PER 11,247 3,311 VEN 1,106 1,167 Note. (a) From International Monetary Fund. Retrieved March 3, 2008, from http://www.imf.org/external/index.htm. (b) From Instituto Nacional de Estadistica y Censos de Argentina and Instituto de Pesquisa Economica Aplicada de Brazil. Retrieved March 6, 2008, from http://www.indec.gov.ar/ and http://www.ipeadata.gov.br. From Bancos Centrales de Chile, Colombia, Mexico, Peru, and Venezuela. Retrieved March 4, 2008, from http://www.bcentral.cl, http://www.banrep.gov.co, www.banxico.org.mx, http://estadisticas.bcrp.gob.pe, and http://www.bcv.org.ve. (c) From Federacion Iberoamericana de Bolsas. Retrieved March 6, 2008, from http://www.fiabnet.org/es/index.asp.g Table 2 Sample Moments of the Distributions of Daily Returns Market (a) M SD Min return ARG 0.000689 0.021548 -0.112907 BRA 0.000665 0.018219 -0.096286 CHI 0.000503 0.006120 -0.038535 COL 0.001157 0.013888 -0.110519 MEX 0.000706 0.013974 -0.082673 PER 0.001131 0.011340 -0.078929 VEN 0.001020 0.016077 -0.206580 Market (a) Max return Skewness Kurtosis ARG 0.161165 0.183990 4.970798 BRA 0.073356 -0.254298 0.943668 CHI 0.027785 -0.396688 3.225253 COL 0.146880 -0.051487 14.830800 MEX 0.070199 -0.116522 2.806384 PER 0.082050 -0.329385 6.379885 VEN 0.131632 -0.560760 24.305530 Note. Skewness refers to a standardized third central sample moment, and kurtosis refers to a standardized fourth central sample moment. (a) Daily observations obtained from Bolsa de Valores de Argentina (http://www.merval.sba.com), Brazil (http://www.bovespa.com), Chile (http://www.bolsadesantiago.com), Colombia (http://www.bvc.com), Mexico (http://www.bmv.com), Peru (http://www.bvl.com), and Venezuela (http://www.bolsadecaracas.com). Table 3 Standardized Skewness and Kurtosis Standardized Market N Skewness skewness ARG 1,978 0.183990 3.340665 BRA 1,982 -0.254298 -4.621894 CHI 1,992 -0.396688 -7.227994 COL 1,944 -0.051487 -0.926767 MEX 2,013 -0.116522 -2.134294 PER 1,995 -0.329385 -6.006197 VEN 1,908 -0.560760 -9.999787 Standardized Market Kurtosis kurtosis ARG 4.970798 45.126711 BRA 0.943668 8.575615 CHI 3.225253 29.383449 COL 14.830800 133.477202 MEX 2.806384 25.701793 PER 6.379885 58.167274 VEN 24.305530 216.714840 Note. Standardized skewness is computed based on zero mean and variance of 6/N; standardized kurtosis is computed based on zero mean and variance of 24/N. Table 4 Normality Tests Shapiro-Wilk Jarque-Bera Market Statistic p Statistic p ARG 0.9416 0.0000 2,035 0.0000 BRA 0.9927 0.0000 94 0.0000 CHI 0.9697 0.0000 910 0.0000 COL 0.8701 0.0000 17,719 0.0000 MEX 0.9680 0.0000 660 0.0000 PER 0.9283 0.0000 3,395 0.0000 VEN 0.8249 0.0000 46,808 0.0000 Pearson chi-square Shapiro-Francia Market Statistic p Statistic p ARG 228 0.0000 0.9402 0.0000 BRA 56 0.0410 0.9921 0.0000 CHI 86 0.0000 0.9687 0.0000 COL 340 0.0000 0.8673 0.0000 MEX 146 0.0000 0.9671 0.0000 PER 271 0.0000 0.9266 0.0000 VEN 447 0.0000 0.8215 0.0000 Kolmogorov-Smirnov Market Statistic p ARG 0.0722 0.0000 BRA 0.0282 0.0009 CHI 0.0388 0.0000 COL 0.0964 0.0000 MEX 0.0522 0.0000 PER 0.0803 0.0000 VEN 0.1077 0.0000 Note. Statistics estimated from stock market returns. Table 5 Probability Discrepancies Between Adjusted Normal Distribution and Empirical Distribution Normal theoretical Stock return interval probability ARG BRA [[R.sup.e] - 5S, [R.sup.e] - 6S] 0.00000 0.00051 0.00050 [[R.sup.e] - 4S, [R.sup.e] - 5S] 0.00003 0.00253 0.00050 [[R.sup.e] - 3S, [R.sup.e] - 4S] 0.00131 0.00657 0.00303 [[R.sup.e] - 2S, [R.sup.e] - 3S] 0.02136 0.01820 0.02472 [[R.sup.e] - S, [R.sup.e] - 2S] 0.13585 0.08544 0.11504 [[R.sup.e], [R.sup.e] - S] 0.34144 0.38170 0.34309 [[R.sup.e], [R.sup.e] + S] 0.34144 0.39687 0.36478 [[R.sup.e] + S, [R.sup.e] + 2S] 0.13585 0.07988 0.12866 [[R.sup.e] + 2S, [R.sup.e] + 3S] 0.02136 0.02073 0.01816 [[R.sup.e] + 3S, [R.sup.e] + 4S] 0.00131 0.00506 0.00151 [[R.sup.e] + 4S, [R.sup.e] + 5S] 0.00003 0.00101 0.00000 [[R.sup.e] + 5S, [R.sup.e] + 6S] 0.00000 0.00101 0.00000 Stock return interval CHI COL MEX [[R.sup.e] - 5S, [R.sup.e] - 6S] 0.00100 0.00154 0.00050 [[R.sup.e] - 4S, [R.sup.e] - 5S] 0.00050 0.00103 0.00149 [[R.sup.e] - 3S, [R.sup.e] - 4S] 0.00703 0.00566 0.00447 [[R.sup.e] - 2S, [R.sup.e] - 3S] 0.01657 0.01646 0.02235 [[R.sup.e] - S, [R.sup.e] - 2S] 0.10643 0.06533 0.09836 [[R.sup.e], [R.sup.e] - S] 0.36245 0.40226 0.35221 [[R.sup.e], [R.sup.e] + S] 0.37550 0.40278 0.40537 [[R.sup.e] + S, [R.sup.e] + 2S] 0.10793 0.08693 0.09041 [[R.sup.e] + 2S, [R.sup.e] + 3S] 0.01857 0.01029 0.01639 [[R.sup.e] + 3S, [R.sup.e] + 4S] 0.00201 0.00257 0.00596 [[R.sup.e] + 4S, [R.sup.e] + 5S] 0.00151 0.00103 0.00248 [[R.sup.e] + 5S, [R.sup.e] + 6S] 0.00000 0.00154 0.00000 Stock return interval PER VEN [[R.sup.e] - 5S, [R.sup.e] - 6S] 0.00050 0.00105 [[R.sup.e] - 4S, [R.sup.e] - 5S] 0.00201 0.00262 [[R.sup.e] - 3S, [R.sup.e] - 4S] 0.00602 0.00105 [[R.sup.e] - 2S, [R.sup.e] - 3S] 0.01855 0.01101 [[R.sup.e] - S, [R.sup.e] - 2S] 0.06667 0.06027 [[R.sup.e], [R.sup.e] - S] 0.42155 0.44759 [[R.sup.e], [R.sup.e] + S] 0.37694 0.39465 [[R.sup.e] + S, [R.sup.e] + 2S] 0.07920 0.05346 [[R.sup.e] + 2S, [R.sup.e] + 3S] 0.01955 0.01625 [[R.sup.e] + 3S, [R.sup.e] + 4S] 0.00602 0.00629 [[R.sup.e] + 4S, [R.sup.e] + 5S] 0.00100 0.00157 [[R.sup.e] + 5S, [R.sup.e] + 6S] 0.00000 0.00105 Note. [R.sup.e] refers to the sample mean, and S refers to the sample's standard deviation of the corresponding daily market return. Table 6 Outliers < -3S < -4S Expected No. Expected No. no. cases cases in no. cases cases in for normal empirical for normal empirical Market distribution data distribution data Negative outliers ARG 2.6703 19 0.0791 6 BRA 2.6757 8 0.0793 2 CHI 2.6892 18 0.0797 4 COL 2.6244 19 0.0778 8 MEX 2.7175 13 0.0805 4 PER 2.6932 20 0.0798 8 VEN 2.5758 11 0.0763 9 Positive outliers ARG 2.6703 15 0.0791 5 BRA 2.6757 3 0.0793 0 CHI 2.6892 7 0.0797 3 COL 2.6244 12 0.0778 7 MEX 2.7175 17 0.0805 5 PER 2.6932 15 0.0798 3 VEN 2.5758 21 0.0763 9 < -5S < -6S Expected No. Expected No. no. cases cases in no. cases cases in for normal empirical for normal empirical Market distribution data distribution data Negative outliers ARG 0.0198 1 0.0192 0 BRA 0.0198 1 0.0193 0 CHI 0.0199 3 0.0194 1 COL 0.0194 6 0.0189 3 MEX 0.0201 1 0.0196 0 PER 0.0199 4 0.0194 3 VEN 0.0191 4 0.0185 2 Positive outliers ARG 0.0198 3 0.0192 1 BRA 0.0198 0 0.0193 0 CHI 0.0199 0 0.0194 0 COL 0.0194 5 0.0189 2 MEX 0.0201 0 0.0196 0 PER 0.0199 1 0.0194 1 VEN 0.0191 6 0.0185 4 Table 7 Sample Moments of the Distributions of Returns for Argentina K = 1 K = 5 K = 20 N 1,978 395 98 M return 0.0006892 0.0034436 0.0141703 SD of returns 0.0215476 0.0495564 0.1133183 Min return -0.1129074 -0.2138530 -0.3678334 Max return 0.1611652 0.2297794 0.5934334 Skewness 0.1839905 -0.0877070 0.4940497 Standardized skewness 3.3406648 -0.7116352 1.9966781 Kurtosis 4.9707985 2.6717777 7.9733762 Standardized kurtosis 45.126711 10.8391008 16.1120082 K = 60 K = 120 N 32 16 M return 0.0440582 0.0881164 SD of returns 0.1970460 0.2811204 Min return -0.4055984 -0.6752064 Max return 0.5757295 0.3976514 Skewness 0.1227302 -1.3036006 Standardized skewness 0.2834332 -2.1287708 Kurtosis 0.9162758 2.3497764 Standardized kurtosis 1.0580242 1.9185844 Note. Statistics estimated from stock market returns. Table 8 Argentina: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.9416 0.0000 2035.00 0.0000 5 0.9662 0.0000 113.73 0.0000 20 0.8712 0.0000 234.27 0.0000 60 0.9771 0.7121 0.55 0.7600 120 0.8839 0.0447 4.87 0.0876 Pearson chi-square Shapiro-Francia K Statistic p Statistic p 1 227.79 0 0.9402 0.0000 5 28.90 0.0675 0.9626 0.0000 20 21.39 0.0186 0.8552 0.0000 60 4.50 0.4799 0.9684 0.3860 120 5.00 0.2873 0.8781 0.0366 Kolmogorov-Smirnov K Statistic p 1 0.0722 0.0000 5 0.0661 0.0003 20 0.1115 0.0044 60 0.1035 0.5168 120 0.1354 0.6025 Note. Statistics estimated from stock market returns. Table 9 Sample Moments of the Distributions of Returns for Brazil K = 1 K = 5 K = 20 N 1,982 396 99 M return 0.0006652 0.0033454 0.0133815 SD of returns 0.0182195 0.0409020 0.0810222 Min return -0.0962865 -0.1381972 -0.1867589 Max return 0.0733561 0.1348822 0.2229557 Skewness -0.2542985 -0.1454410 -0.3646627 Standardized skewness -4.6218943 -1.1815680 -1.4812667 Kurtosis 0.9436676 0.6154016 0.0655374 Standardized kurtosis 8.5756148 2.4997733 0.1331071 K = 60 K = 120 N 33 16 M return 0.0401444 0.0772821 SD of returns 0.1188910 0.1639827 Min return -0.2561701 -0.1674060 Max return 0.2625434 0.3369533 Skewness -0.3306758 -0.0336668 Standardized skewness -0.7755035 -0.0549776 Kurtosis 0.1215786 -1.5151553 Standardized kurtosis 0.1425636 -1.2371191 Note. Statistics estimated from stock market returns. Table 10 Brazil: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.9927 0.0000 94.04 0.0000 5 0.9932 0.0705 7.18 0.0276 20 0.9812 0.1681 2.13 0.3451 60 0.9787 0.7461 0.55 0.7579 120 0.9185 0.1594 1.37 0.5031 Pearson chi-square Shapiro-Francia K Statistic p Statistic p 1 55.62 0.0410 0.9921 0.0000 5 17.00 0.5899 0.9924 0.0420 20 9.60 0.4766 0.9818 0.1645 60 5.45 0.4870 0.9787 0.6545 120 11.13 0.0252 0.9356 0.2544 Kolmogorov-Smirnov K Statistic p 1 0.0282 0.0009 5 0.0412 0.1048 20 0.0820 0.0971 60 0.0910 0.6965 120 0.1734 0.2229 Note. Statistics estimated from stock market returns. Table 11 Sample Moments of the Distributions of Returns for Chile K = 1 K = 5 N 1,992 398 M return 0.0005030 0.0025178 SD of returns 0.0061202 0.0173836 Min return -0.0385347 -0.0599934 Max return 0.0277847 0.0646196 Skewness -0.3966877 -0.3445022 Standardized skewness -7.2279945 -2.8058080 Kurtosis 3.2252526 1.5865871 Standardized kurtosis 29.3834490 6.4610013 K = 20 K = 60 K = 120 N 99 33 16 M return 0.0103292 0.0309875 0.0634268 SD of returns 0.0386314 0.0598168 0.0970733 Min return -0.1358573 -0.0727310 -0.0648024 Max return 0.0903600 0.1587032 0.2436628 Skewness -0.4742471 0.2925534 0.3678472 Standardized skewness -1.9264008 0.6860985 0.6006919 Kurtosis 1.1014442 -0.5780826 -0.8622138 Standardized kurtosis 2.2370437 -0.6778620 -0.7039946 Note. Statistics estimated from stock market returns. Table 12 Chile: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.9697 0.0000 909.70 0.0000 5 0.9774 0.0000 47.74 0.0000 20 0.9802 0.1421 7.61 0.0222 60 0.9751 0.6308 1.05 0.5925 120 0.9381 0.3261 0.92 0.6314 Pearson chi-square Shapiro-Francia K Statistic p Statistic p 1 86.37 0.0000 0.9687 0.0000 5 31.06 0.0398 0.9760 0.0000 20 4.87 0.8998 0.9774 0.0790 60 3.27 0.7739 0.9820 0.7599 120 6.75 0.1497 0.9530 0.4587 Kolmogorov-Smirnov K Statistic p 1 0.0388 0.0000 5 0.0481 0.0276 20 0.0555 0.6370 60 0.0925 0.6728 120 0.1388 0.5625 Note. Statistics estimated from stock market returns. Table 13 Sample Moments of the Distributions of Returns for Colombia K = 1 K = 5 K = 20 N 1,944 388 97 M return 0.0011567 0.0057948 0.0231791 SD of returns 0.0138876 0.0364820 0.0846745 Min return -0.1105192 -0.2988605 -0.2936002 Max return 0.1468797 0.1365904 0.2392459 Skewness -0.0514870 -1.6011630 -0.4344733 Standardized skewness -0.9267667 -12.875846 -1.7469211 Kurtosis 14.8308000 13.869213 1.4635376 Standardized kurtosis 133.4772024 55.765047 2.9422808 K = 60 K = 120 N 32 16 M return 0.0723743 0.1447487 SD of returns 0.1320623 0.1896683 Min return -0.2322182 -0.2704352 Max return 0.3332463 0.5680246 Skewness 0.1401023 -0.0185203 Standardized skewness 0.3235524 -0.0302436 Kurtosis -0.0269279 1.4105894 Standardized kurtosis -0.0310937 1.1517414 Note. Statistics estimated from stock market returns. Table 14 Colombia: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.8701 0.0000 17,719.27 0.0000 5 0.8862 0.0000 3,188.08 0.0000 20 0.9813 0.1814 10.09 0.0065 60 0.9702 0.5061 0.1509 0.9273 120 0.9699 0.8367 0.29 0.8669 Pearsonn chi-square Shapiro-Francia K Statistic p Statistic p 1 340.15 0.0000 0.8673 0.0000 5 49.28 0.0002 0.8791 0.0000 20 11.69 0.3063 0.9760 0.0681 60 7.00 0.2206 0.9704 0.4344 120 3.25 0.5169 0.9510 0.4286 Kolmogorov-Smirnov K Statistic p 1 0.0964 0.0000 5 0.0868 0.0000 20 0.0619 0.4790 60 0.1112 0.4002 120 0.1172 0.8055 Note. Statistics estimated from stock market returns. Table 15 Sample Moments of the Distributions of Returns for Mexico K = 1 K = 5 K = 20 N 2,013 402 100 M return 0.0007061 0.0035746 0.0147557 SD of returns 0.0139744 0.0344861 0.0675184 Min return -0.0826735 -0.1563092 -0.1870277 Max return 0.0701992 0.1591427 0.1282246 Skewness -0.1165220 -0.5229036 -1.2101192 Standardized skewness -2.1342943 -4.2801502 -4.9402909 Kurtosis 2.8063844 2.8989659 1.1759190 Standardized kurtosis 25.7017933 11.8645294 2.4003346 K = 60 K = 120 N 33 16 M return 0.0426807 0.0850310 SD of returns 0.1056983 0.1570158 Min return -0.2364434 -0.1897873 Max return 0.1630990 0.2899391 Skewness -0.9981216 -0.5494813 Standardized skewness -2.3408026 -0.8972993 Kurtosis 0.3141320 -1.0994583 Standardized kurtosis 0.3683524 -0.8977040 Note. Statistics estimated from stock market returns. Table 16 Mexico: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.9680 0.0000 660.40 0.0000 5 0.9583 0.0000 154.06 0.0000 20 0.8934 0.0000 28.35 0.0000 60 0.8921 0.0033 5.00 0.0822 120 0.9034 0.0912 1.52 0.4676 Pearson chi-square Shapiro-Francia K Statistic p Statistic p 1 146.44 0.0000 0.9671 0.0000 5 48.27 0.0004 0.9546 0.0000 20 38.58 0.0000 0.8945 0.0000 60 15.27 0.0182 0.8973 0.0062 120 5.00 0.2873 0.9195 0.1450 Kolmogorov-Smirnov K Statistic p 1 0.0522 0.0000 5 0.0752 0.0000 20 0.1473 0.0000 60 0.1606 0.0303 120 0.2006 0.0846 Note. Statistics estimated from stock market returns. Table 17 Sample Moments of the Distributions of Returns for Peru K = 1 K = 5 K = 20 N 1,995 399 99 M return 0.0011310 0.0056548 0.0234562 SD of returns 0.0113402 0.0329057 0.0767978 Min return -0.0789286 -0.1573820 -0.1660220 Max return 0.0820498 0.1322477 0.3086597 Skewness -0.3293849 -0.3875031 0.7385187 Standardized skewness -6.0061970 -3.1599921 2.9998772 Kurtosis 6.3798855 4.5076570 2.0287995 Standardized kurtosis 58.167274 18.3794150 4.1205113 K = 60 K = 120 N 33 16 M return 0.0703685 0.1454678 SD of returns 0.1371113 0.2173732 Min return -0.1546970 -0.2226098 Max return 0.3181100 0.5470086 Skewness 0.0911673 -0.2074088 Standardized skewness 0.2138062 -0.3386972 Kurtosis -1.0028769 0.0023149 Standardized kurtosis -1.1759774 0.0018901 Note. Statistics estimated from stock market returns. Table 18 Peru: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic p Statistic p 1 0.9283 0.0000 3,395.04 0.0000 5 0.9283 0.0000 337.11 0.0000 20 0.9588 0.0035 23.12 0.0000 60 0.9653 0.3631 1.51 0.4698 120 0.9303 0.2460 0.18 0.9158 Pearson chi-square Shapiro-Francia K Statistic p Statistic p 1 270.71 0.0000 0.9266 0.0000 5 66.53 0.0000 0.9246 0.0000 20 11.96 0.2878 0.9536 0.0023 60 3.27 0.7739 0.9779 0.6282 120 5.88 0.2087 0.9333 0.2343 Kolmogorov-Smirnov K Statistic p 1 0.0803 0.0000 5 0.0962 0.0000 20 0.0741 0.1998 60 0.0799 0.8561 120 0.1825 0.1650 Note. Statistics estimated from stock market returns. Table 19 Sample Moments of the Distributions of Returns for Venezuela K = 1 K = 5 K = 20 N 1,908 381 95 M return 0.0010196 0.0050846 0.0200937 SD of returns 0.0160774 0.0409763 0.0844795 Min return -0.2065804 -0.2638650 -0.2158451 Max return 0.1316321 0.2183877 0.3892106 Skewness -0.5607602 0.3010843 0.8703767 Standardized skewness -9.9997870 2.3992474 3.4633264 Kurtosis 24.3055300 7.7321864 3.4951914 Standardized kurtosis 216.7148400 30.8076930 6.9538793 K = 60 K = 120 N 31 15 M return 0.0623839 0.1360172 SD of returns 0.1580321 0.2722082 Min return -0.2427682 -0.4376394 Max return 0.4529515 0.6567241 Skewness 0.3536995 0.0675908 Standardized skewness 0.8039696 0.1068704 Kurtosis 0.3932781 0.7954750 Standardized kurtosis 0.4469665 0.6288782 Note. Statistics estimated from stock market returns. Table 20 Venezuela: Normality Tests Shapiro-Wilk Jarque-Bera K Statistic P Statistic p 1 1 0 46,807.75 0.00 5 1 0.00% 92637.00% 0 20 1 0.06% 5346.00% 0 60 1 77.10% 61.00% 1 120 1 74.09% 3.00% 1 Pearson chi-square Shapiro-Francia K Statistic p Statistic P 1 447.08 0 0.8215 0 5 62 0 0.8965 0 20 22 0.0166 0.9359 0.0003 60 3 0.6504 0.9768 0.6273 120 4 0.2839 0.952 0.4734 Kolmogorov-Smirnov K Statistic p 1 0.1077 0 5 0.1016 0 20 0.1014 0.0174 60 0.1242 0.2577 120 0.1179 0.8316 Note. Statistics estimated from stock market returns.
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|Author:||Chion, Sergio J.; Veliz C., Carlos N.|
|Publication:||Journal of CENTRUM Cathedra|
|Date:||Sep 1, 2008|
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