Normal distribution of returns of 65 stock exchange indexes

In the first part of the paper the hypothesis of normality for returns of 65 equity indexes was verified. The statistical hypothesis for each of analyzed indexes was verified for the following time intervals: daily, weekly, monthly, quarterly and yearly. For each of the analyzed indices the following rates of return were calculated (daily rates of return): close-close, overnight, open-open and open-close. Verification of statistical hypotheses was conducted with the use of the following five statistical tests: Jarque-Bera, Lilliefors, Cramer von Mises, Watson and Anderson-Darling.

In the second part, the hypothesis of the normality of daily returns for 6 indexes: CAC40, DAX, DJIA, FTSE250, NIKKEI and S&P 500 was verified in the annual time horizons, i.e. for the following years 2013-2016. For the DJIA index also the normality of daily returns in 28 upward and downward waves was verified with the use of the following tests: Jarque-Bera, Kolmorgow-Smirnow, Lilliefors, Cramer von Mises, Watson and Anderson-Darling.

In the third part of the paper, with the use of the parameter p, a stock index ranking was created, due to the possibility of approximating the distribution of index returns with a normal distribution for the time horizon of K = 30, K = 126 and K = 252 sessions. In the process of ranking, the following tests: Jarque-Bera, Shapiro-Wilk and D'Agostino-Pearson were implemented.

The paper shows that the distribution of the remaining daily returns, e.g. O-O, C-O and overnight, calculated for the analyzed equity indexes, does not follow the path of a normal distribution. It has been proven that the distribution of returns can be normal only in given time intervals. Time intervals can be set up as individual years or up and down waves. The results can be considered as a voice in an ongoing discussion about the distribution of returns and thus the efficiency of financial markets, to be used in the process of building investment strategies.