Research Journal of Management Sciences _____________________________________________ISSN 2319–1171 Vol. 2(11), 1-15, November (2013) Res. J. Management Sci. International Science Congress Association 1 Modeling the Selection of Returns Distribution of G7 Countries G.S. David Sam Jayakumar and Sulthan A. Jamal Institute of Management, Tiruchirappalli, South India, INDIA Available online at: www.isca.in, www.isca.me Received 23rd August 2013, revised 9th September 2013, accepted 26th October 2013 AbstractThe purpose of the study is to identify the statistical distribution that is followed by the Indices returns of G7 countries. Canada is one of the members of the G7 countries but the data is insufficient and so it is not considered for the study. The closing values of indices were collected for selected countries from July 2003 to February 2013. The general assumption is that the stock returns are normally distributed. Using a statistical software 11 unbounded distributions was fitted for all the selected indices. The results show the returns follow different distributions that vary between countries. Keywords: Unbounded distribution, G7 countries, Indices returns. Introduction The assumption that stock returns are normally distributed is widely used, implicitly or explicitly, in theoretical finance. The fitting of probability distributions to financial data is a statistical subject with a long tradition in both actuarial and financial literature. It was Louis Bachelier who used a stochastic approach to model nancial time series for the rst time. In 1973, Fischer Black and Myron Scholes published their famous work where they presented a model for pricing European options. They assumed that a price of an asset can be described by a geometric Brownian motion. However, Mandelbrotshowed the behaviour of real markets differs from the Brownian property, since the price returns form a truncated Levy distribution4,5. As a result of this observation many non-Gaussian models were introduced by Mantegna and Stanleyand Bouchaud. Another divergence from the Gaussian behaviour is an autocorrelation in nancial systems. Empirical studies show that the autocorrelation function of the stock market time series decays exponentially with a characteristic time of a few minutes, while the absolute values of the autocorrelation of prices decay more slowly, as a power law function, which leads to a volatility clustering8, 9. Jansen and De Vries10, used extremes to investigate the fatness of the distribution tails. In this study, the authors fit 11 unbounded distributions to selected countries to understand the statistical distribution followed by the country’s index returns. Methodology In case of time series analysis selecting an appropriate distribution is very important to get a meaning full result from the analysis however the nature of data differ from each other this necessary to find its nature of distribution so this study was made to find distribution G7 countries. The closing values of index for G7 countries from July 2003 to February 2013 was collected from yahoo finance website. After secondary data collection is over the returns were analyzed with help of math values easy fit 5.5.the analysis done in different stage in first stage in descriptive statistic identify for the returns of G7 countries. In next stage eleven unbounded distribution for fitted for all six countries. Canada is one of the members of the G7 countries but the data is insufficient and so it is not considered for the study. Theoretical frame work of unbounded probability distribution: Gumbel min Distribution: In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Such a distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The cumulative distribution function of the Gumbel distribution is ()/ (;,)Fxe mb mb--The mode is µ, while the median is ln(ln2) mb and the mean is given by()EX mgb =+ Where g =Euler-Mascheroni constant 0.5772 The standard deviation is /6 bpCauchy Distribution: The Cauchy distribution, named after Augustine, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz (ian) function, or Breit – Wigner distribution. The simplest Cauchy distribution is called the standard Cauchy distribution. It has the distribution of a random variable that is the ratio of two independent standard normal random variables. This has the probability density function Research Journal of Management Sciences _________________________________________________________ISSN 2319–1171Vol. 2(11), 1-15, November (2013) Res. J. Management Sci.International Science Congress Association 2 2 (;0,1) (1) fx x Its cumulative distribution function has the shape of an arctangent function arctan (): 11 (;0,1)arctan() 2 Fxx x =+ Johnson Su Distribution: The Johnson SU distribution is a four-parameter family of probability distributions first investigated by Johnson in 1949.It is closely related to the normal distribution. Generation of random variables: Let be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson SU random variables can be generated from as follows: sinh() xux lg  =F-+   where is the cumulative distribution function of the normal distribution. Normal distribution: In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution, defined by the formula 2 2 () ()fxe m spThe parameter in this formula is the mean or expectation of the distribution (and also its median and mode). The parameter is its standard deviation; its variance is therefore . A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate. Logistic Distribution: In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feed forward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). Probability density function: The probability density function (pdf) of the logistic distribution is given by: (;,)sec42exfxsh ss se m m-----  ==    Because the pdf can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution. Laplace Distribution: In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential (with an additional location parameter) spliced together back-to-back, but the term double exponential distribution is also sometimes used to refer to the Gumbel distribution. The difference between two independent exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution. Error Function: In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as: () tdt erfxe  The complementary error function, denoted erfc, is defined as 2 ()1() () tdt x erfxerfx erfxe =- =  The imaginary error function, denoted erfi, is defined as ()() erfzierfiz =- When the error function is evaluated for arbitrary complex arguments , the resulting complex error function is usually discussed in scaled form as the Faddeeva function: ()() z zeerfciz w- =- Hyperbolic secant distribution: In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the inverse hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution. A random variable follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation: ()sec 22 fxhx p  =   Where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) of the standard distribution is Research Journal of Management Sciences _________________________________________________________ISSN 2319–1171Vol. 2(11), 1-15, November (2013) Res. J. Management Sci.International Science Congress Association 3 11()arctansec22()arctanexp fxhx fxx \n  =+   \r \n \rwhere "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is () ()sinhcot()lntan Fparp Fpp=- \n \r \n \rwhere "arsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution. Student's -distribution: In probability and statistics, Student's t-distribution (or simply the t-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It plays a role in a number of widely used statistical analyses, including the Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's t-distribution also arises in the Bayesian analysis of data from a normal family. If we take k samples from a normal distribution with fixed unknown mean and variance, and if we compute the sample mean and sample variance for these k samples, then the distribution (for k) can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term. In this way the t-distribution can be used to estimate how likely it is that the true mean lies in any given range. The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's t-distribution is a special case of the generalised hyperbolic distribution. Results and discussion Table-1 visualises the result of descriptive that statistics shows the Minimum, Mean, Maximum, Standard Deviation, Covariance, The average return of FTSE100 is higher when comparative to the average return of other G7 countries. The result of Standard Deviation of shows Italy has higher volatility when compared to other selected countries. Table-2 visualizes the parameter of different unbounded distribution fitted for Nikkei. In table-3 the goodness of fit is shown for different distribution fitted for returns of Nikkei. The Johnson Su distribution has the best fit when compared to other unbounded distributions. Table-4 shows the parameter of different unbounded distribution fitted for Nikkei. In table-5 the goodness of fit is shown for different distribution fitted for returns of NASDAQ. The Johnson Su distribution has the best fit in compared to other unbounded distributions. Table-6 shows visualize the parameter of different unbounded distribution fitted for Nikkei. In of table-7 the goodness of fit is shown for different distribution fitted for returns of CAC 40. The Gumbel Min distribution has the best fit in compared to other unbounded distributions. Table-8 shows visualize the parameter of different unbounded distribution fitted for Nikkei. In table-9 the goodness of fit is shown for different distribution fitted for returns of DBE. The Johnson Su distribution has the best fit in compared to other unbounded distributions. Table-10 shows visualize the parameter of different unbounded distribution fitted for Nikkei. In of table-11 the goodness of fit is shown for different distribution fitted for returns of FTSE.MIB. The Cauchy distribution has the best fit in compared to other unbounded distributions. Table-12 shows visualize the parameter of different unbounded distribution fitted for Nikkei. In of table-13 the goodness of fit is shown for different distribution fitted for returns of FTSE 100. The Gumbel min distribution has the best fit in compared to other unbounded distributions. Table-1 Descriptive Statistics Country Index Minimum Mean Maximum Standard Deviation C.V NIKKEI -23.8269 0.3764 12.8499 5.7359 15.2357 FTSE.MIB -16.3063 1.0956 155.130 15.5759 14.2159 NASDAQ -17.7319 0.7133 12.3454 5.2211 7.3194 FTSE100 -13.0238 0.4710 8.4533 3.9129 8.3069 DBE -19.1921 0.9093 16.7621 5.4483 5.9914 CAC40 -13.5173 0.2808 12.5567 4.8376 17.2275 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Maximum Likelihood Estimates Unbounded distribution  Cauchy 3.0377 0.76695 Error 5.7359 0.37648 Error Function - Gumbel Max 4.4723 2.205 Gumbel Min 4.4723 2.9579 Hypersecant 5.7359 0.37648 Johnson SU - Laplace - 0.37648 Logistic 3.1624 0.37648 Normal 5.7359 0.37648 Student's t - Unbounded distribution Kolmogorov Smirnov Statistic Cauchy 0.08699 Error 0.06639 Error Function 0.11376 Gumbel Max 0.14915 Gumbel Min 0.09195 Hypersecant 0.06608 Johnson SU 0.06153 Laplace 0.08181 Logistic 0.07602 Normal 0.08811 Student's t 0.30987 Figure-1 Figure showing _______________________ __________ ________________________ Association Table-2 Estimates of Unbounded Distribution Parameters for NIKKEI Parameters(Scale, Shape, Allocation ) µ K 0.76695 - - - - 0.37648 1.1516 - - - - - 0.12328 - - 2.205 - - - - 2.9579 - - - - 0.37648 - - - - - - - 9.9061 2.1882 6.0448 0.37648 - - - - 0.37648 - - - - 0.37648 - - - - - - - - - Table-3 Goodness of Fit – Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.08699 6 1.186 9 0.06639 3 0.47786 4 0.11376 9 1.1236 8 0.14915 10 4.6355 10 0.09195 8 0.98342 7 0.06608 2 0.43631 3 0.06153 1 0.24798 1 0.08181 5 0.71043 6 0.07602 4 0.39937 2 0.08811 7 0.60059 5 0.30987 11 39.103 11 Figure-2 showing Probability density function for NIKKEI ________________________ ISSN 2319–1171 Res. J. Management Sci. 4 NIKKEI    - - - - - - - - - - - - - - - - - - 6.0448 1.0832 - - - - - - - - - - - - 2 Chi-Squared Statistic Rank 2.2751 1 3.1663 4 8.857 9 10.199 10 4.329 6 2.5044 3 4.4021 7 4.1699 5 2.2806 2 5.7298 8 111.66 11 Figure-3 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-4 Figure-7 Figure Figure showing _______________________ __________ ________________________ Association Figure-5 Figure-8 Figure -10 Figure-11 showing Probability density function for NIKKEI ________________________ ISSN 2319–1171 Res. J. Management Sci. 5 Figure-6 Figure-9 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Maximum Likelihood Estimates Unbounded distribution  Cauchy 3.0862 1.4813 Error 5.2211 0.71332 Error Function - Gumbel Max 4.0709 1.6365 Gumbel Min 4.0709 3.0631 Hypersecant 5.2211 0.71332 Johnson SU - Laplace - 0.71332 Logistic 2.8786 0.71332 Normal 5.2211 0.71332 Student's t - Distribution Kolmogorov Smirnov Statistic Cauchy 0.10386 Error 0.08828 Error Function 0.11907 Gumbel Max 0.13241 Gumbel Min 0.07362 Hypersecant 0.10343 Johnson SU 0.05035 Laplace 0.1306 Logistic 0.08859 Normal 0.06924 Student's t 0.36869 Figure-12 Figure showing _______________________ __________ ________________________ Association Table-4 Estimates of Unbounded Distribution Parameters for NASDAQ Parameters(Scale, Shape, Allocation ) µ K 1.4813 - - - - 0.71332 1.5355 - - - - - 0.13543 - - 1.6365 - - - - 3.0631 - - - - 0.71332 - - - - - - - 14.865 3.7332 0.71332 - - - - 0.71332 - - - - 0.71332 - - - - - - - - - Table-5 Goodness of Fit – Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.10386 7 2.1934 8 0.08828 4 0.7959 4 0.11907 8 2.3751 9 0.13241 10 4.8265 10 0.07362 3 0.74403 2 0.10343 6 1.0569 6 0.05035 1 0.32622 1 0.1306 9 1.8301 7 0.08859 5 0.80796 5 0.06924 2 0.75442 3 0.36869 11 47.473 11 Figure-13 showing Probability density function for NASDAQ ________________________ ISSN 2319–1171 Res. J. Management Sci. 6 NASDAQ Parameters(Scale, Shape, Allocation )    - - - - - - - - - - - - - - - - - - 12.33 2.5912 - - 0.27086 - - - - - - - - - 2 Chi-Squared Statistic Rank 9.3559 4 9.5098 5 10.789 7 19.346 10 3.9459 1 12.283 8 6.9946 2 18.511 9 8.5055 3 10.761 6 72.473 11 Figure-14 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-15 Figure-18 Figure Figure showing _______________________ __________ ________________________ Association Figure-16 Figure-19 Figure -21 Figure-22 showing Probability density function for NASDAQ ________________________ ISSN 2319–1171 Res. J. Management Sci. 7 Figure-17 Figure-20 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Maximum Likelihood Estimates Unbounded distribution  Cauchy 2.6209 Error 4.8376 Error Function - Gumbel Max 3.7719 Gumbel Min 3.7719 Hypersecant 4.8376 Johnson SU - Laplace - Logistic 2.6671 Normal 4.8376 Student's t - Distribution Kolmogorov Smirnov Statistic Cauchy 0.10744 Error 0.10629 Error Function 0.1235 Gumbel Max 0.17075 Gumbel Min 0.04252 Hypersecant 0.11278 Laplace 0.13859 Logistic 0.10721 Normal 0.10062 Student's t 0.3533 Johnson SU Figure-23 Figure showing _______________________ __________ ________________________ Association Table-6 Estimates of Unbounded Distribution Parameters for Parameters(Scale, Shape, Allocation) µ K   1.4276 - - - 0.28081 1.6357 - - - - 0.14617 - 1.8964 - - - 2.458 - - - 0.28081 - - - - - - - 0.28081 - - 0.29234 0.28081 - - - 0.28081 - - - - - - - Table-7 Goodness of Fit – Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.10744 5 2.6428 8 0.10629 3 1.2378 2 0.1235 7 1.8018 6 0.17075 9 6.6645 9 0.04252 1 0.31413 1 0.11278 6 1.4884 5 0.13859 8 2.3391 7 0.10721 4 1.2507 3 0.10062 2 1.2632 4 0.3533 10 42.053 10 No fit Figure-24 showing Probability density function for CAC40 ________________________ ISSN 2319–1171 Res. J. Management Sci. 8 CAC40 Allocation)   - - - - - - - - - - - - - - - - - - - - - 2 Chi-Squared Statistic Rank 11.786 4 10.334 3 12.005 5 23.917 9 2.3208 1 12.396 6 15.11 8 9.7261 2 14.156 7 158.88 10 Figure-25 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-26 Figure-29 Figure showing _______________________ __________ ________________________ Association Figure-27 Figure-30 Figure-32 showing Probability density function for CAC40 ________________________ ISSN 2319–1171 Res. J. Management Sci. 9 Figure-28 Figure-31 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Maximum Likelihood Unbounded distribution  Cauchy 2.575 1.9164 Error 5.4483 0.90935 Error Function - Gumbel Max 4.248 1.5427 Gumbel Min 4.248 3.3614 Hypersecant 5.4483 0.90935 Johnson SU - Laplace - 0.90935 Logistic 3.0038 0.90935 Normal 5.4483 0.90935 Student's t - Distribution Kolmogorov Smirnov Statistic Cauchy 0.09862 Error 0.12871 Error Function 0.16014 Gumbel Max 0.16463 Gumbel Min 0.09951 Hypersecant 0.11531 Johnson SU 0.06951 Laplace 0.14006 Logistic 0.10665 Normal 0.09623 Student's t 0.42517 Figure-33 Figure showing _______________________ __________ ________________________ Association Table-8 Likelihood Estimates of Unbounded Distribution Parameters for Parameters(Scale, Shape, Allocation ) µ K    1.9164 - - - - 0.90935 1.1355 - - - - - 0.12978 - - 1.5427 - - - - 3.3614 - - - - 0.90935 - - - - - - - 9.0834 2.0531 5.0713 0.90935 - - 0.25957 - 0.90935 - - - - 0.90935 - - - - - - - - - Table-9 Goodness of Fit - Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.09862 3 1.9678 8 0.12871 7 1.1639 4 0.16014 9 4.0859 9 0.16463 10 6.3349 10 0.09951 4 1.2954 5 0.11531 6 1.0263 2 0.06951 1 0.5883 1 0.14006 8 1.3676 6 0.10665 5 1.1207 3 0.09623 2 1.632 7 0.42517 11 51.854 11 Figure-34 showing Probability density function for DBE ________________________ ISSN 2319–1171 Res. J. Management Sci. 10 for DBE Parameters(Scale, Shape, Allocation )    - - - - - - - - - - - - - - - - - - 5.0713 0.81402 - - - - - - - - - - - - 2 Chi-Squared Statistic Rank 13.255 5 13.067 4 14.114 6 17.242 9 4.8304 1 15.075 7 8.2325 2 8.5474 3 16.345 8 20.331 10 181.14 11 Figure-35 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-36 Figure-39 Figure showing _______________________ __________ ________________________ Association Figure-37 Figure-40 Figure-42 showing Probability density function for DBE ________________________ ISSN 2319–1171 Res. J. Management Sci. 11 Figure-38 Figure-41 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Maximum Likelihood Estimates Unbounded distribution Parameters(Scale, Shape, Allocation )  Cauchy 3.3521 Error 15.576 Error Functio n - Gumbel Max 12.144 Gumbel Min 12.144 Hypersecant 15.576 Johnson SU - Laplace - Logistic 8.5875 Normal 15.576 Student's t - Distribution Kolmogorov Smirnov Statistic Cauchy 0.09227 Error 0.22488 Error Function 0.24926 Gumbel Max 0.24089 Gumbel Min 0.33484 Hypersecant 0.25167 Laplace 0.22488 Logistic 0.26191 Normal 0.27445 Student's t 0.33385 Johnson SU Figure-43 Figure showing _______________________ __________ ________________________ Association Table-10 Estimates Of Unbounded Distribution Parameters for FTSE.MIB Parameters(Scale, Shape, Allocation ) µ K   0.69134 - - - 1.0957 1.0 - - - - 0.0454 - 5.9143 - - - 8.1057 - - - 1.0957 - - - - - - - 1.0957 - - 0.09079 1.0957 - - - 1.0957 - - - - - - - Table-11 Goodness of Fit – Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.09227 1 1.9327 1 0.22488 3 9.7473 3 0.24926 5 17.539 7 0.24089 4 14.112 6 0.33484 10 22.804 9 0.25167 6 12.412 4 0.22488 2 9.7473 2 0.26191 7 13.856 5 0.27445 8 17.883 8 0.33385 9 47.772 10 No fit Figure-44 showing Probability density function for FTSE.MIB ________________________ ISSN 2319–1171 Res. J. Management Sci. 12 FTSE.MIB   - - - - - - - - - - - - - - - - - - - - - 2 Chi-Squared Statistic Rank 8.6511 1 59.192 3 119.8 8 96.652 6 N/ A 84.805 4 59.192 2 94.421 5 119.26 7 178.73 9 Figure-45 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-46 Figure-49 Figure showing Maximum Likelihood Estimates Unbounded distribution  Cauchy 2.0769 1.1541 Error 3.9129 0.47104 Error Function - Gumbel Max 3.0509 Gumbel Min 3.0509 2.2321 Hypersecant 3.9129 0.47104 Johnson SU - Laplace - 0.47104 Logistic 2.1573 0.47104 Normal 3.9129 0.47104 Student's t - _______________________ __________ ________________________ Association Figure-47 Figure-50 showing Probability density function for FTSE.MIBTable-12 Estimates Of Unbounded Distribution Parameters for FTSE100 Parameters(Scale, Shape, Allocation ) µ K    1.1541 - - - - 0.47104 - - - - - - 0.18071 - - 1.29 - - - - 2.2321 - - - - 0.47104 - - - - - - - 8.2544 3.579 11.318 0.47104 - - 0.36142 - 0.47104 - - - - 0.47104 - - - - - - - - - ________________________ ISSN 2319–1171 Res. J. Management Sci. 13 Figure-48 Figure-51 FTSE100 Parameters(Scale, Shape, Allocation )    - - - - - - - - - - - - - - - - - - 11.318 3.7802 - - - - - - - - - - - - 2 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Distribution Kolmogorov Smirnov Statistic Cauchy 0.08824 Error 0.08694 Error Function 0.12268 Gumbel Max 0.14534 Gumbel Min 0.0741 Hypersecant 0.0942 Johnson SU 0.07632 Laplace 0.12141 Logistic 0.07938 Normal 0.07481 Student's t 0.31599 Figure-52 Figure-55 Figure showing _______________________ __________ ________________________ Association Table-13Goodness of Fit – Summary Kolmogorov Smirnov Anderson Darling Statistic Rank Statistic Rank 0.08824 6 1.8974 8 0.08694 5 0.93782 4 0.12268 9 2.6473 9 0.14534 10 6.1246 10 0.0741 1 0.86617 2 0.0942 7 0.95409 5 0.07632 3 0.49845 1 0.12141 8 1.4745 7 0.07938 4 0.90449 3 0.07481 2 1.1601 6 0.31599 11 28.959 11 Figure-53 Figure-56 showing Probability density function for FTSE 100 ________________________ ISSN 2319–1171 Res. J. Management Sci. 14 Chi-Squared Statistic Rank 14.212 7 13.818 6 15.35 8 25.241 10 1.9334 1 11.286 5 2.2823 2 17.302 9 8.6752 4 8.6119 3 106.53 11 Figure-54 Figure-57 Research Journal of Management Sciences _______________________ Vol. 2(11), 1-15, November (2013) International Science Congress Association Figure-58 Figure-61 Figure showing Probability density function Conclusion From the above made detailed analysis for the the countries namely U.S, U.K, GERMAN, ITALY, JAPAN John Su distributions is well suited index NIKKEI, NASDAQ, DBE. While CAC40 follows Gumbel min distribution and finally, follows Cauchy distributions. When the analysis using the above mentioned unbounded distribution index then there is a good probability for achieving result. As the returns of the indices were considered so it will be appropriate to use unbounded distributions for the study. _______________________ __________ ________________________ Association Figure-59 function for FTSE 100 the selected index of ITALY, FRANCE, suited for the following CAC40 and FTSE 100 finally, FTSE MIB analysis carried out distribution for particular achieving an accurate considered for analysis distributions alone References 1. Bachelier L., Theorie de la speculation, Super,111117, 2186 (1890)2. Black F. and Scholes M., The pricing of options and corporate liabilities, The journal of political economy 3. Bouchaud J.P. and Potters M., derivative pricing: from statistical physics to risk management , Cambridge University Press 4. Jansen D.W. and De Vries C.G., on the frequency of large stock returns: Putting booms and busts into perspective, review of eco nomics and statistics 5. Krawiecki A., Ho\nyst J.A. 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