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2018 | 34 | 5-38

Article title

Testing 65 equity indexes for normal distribution of returns


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Aim/purpose – The primary aim of the paper is to verify the hypothesis on the normal distributions of 65 stock index returns, while the secondary aims are to examine normal distributions for specific years (for six indexes) and for bull and bear markets (for DJIA), demonstrate that the distribution of rates of return for individual indexes can be normal in short time intervals, and then rank analyzed indexes according to the proximity of the distribution of their rates of return to the normal distribution. Design/methodology/approach – The research sample consists of the value of 65 stock indexes from various time intervals. The sample includes both developed markets and emerging markets. The following rates of return were tested for the normality of the rat e of return distribution: close -close, open -open, open -close and overnight, which were calculated for daily, weekly, monthly, quarterly and yearly data. Statistical tests of di f- ferent properties and forces were used: Jarque – Bera (JB), Lilliefors (L), Crame r von Mises (CVM), Watson (W), Anderson – Darling (AD). In the case of six indexes of d e- veloped markets (DJIA, SP500, DAX, CAC40, FTSE250 and NIKKEI225), normality tests of rates distribution were calculated for individual years 2013 -2016 (daily data). In case of the DJIA index, the normality tests of the distribution of returns for individual bull and bear markets were analyzed (daily data, rates of return close -close). In the last part of the paper the analyzed indexes were ranked due to the convergence of their return to normal distribution with the use of the following tests: Jarque – Bera, Shapiro – Wilk and D’Agostino -Pearson. Findings – The distribution of daily and weekly returns of equity indexes is not a normal distribution for all analyzed rates of ret urn. For quarterly and annual data compression the smallest number when there were no reasons to reject the null hypothesis was o b- served for overnight returns compared to close -close, open -close and open -open returns. For the daily, weekly and monthly over night rates of return, the null hypothesis was rejected for all analyzed indexes. The fo llowing general conclusion can be formulated: the higher the data compression (from dail y to yearly), the fewer rejections of H 0 hy- pothesis. The distribution of daily returns can be normal only in given (rather short) time intervals, e.g., particular years or up or do wn waves (bull and bear markets). The posi- tion of the index in the ranking is not depende nt on the date of its first publication, and hence on the number of rates of return possible to calculate for analyzed index, but only on the distribution of its rates of return. Research implications/limitations – The main limitations of the obtained results are different time horizons of each of the analyzed indexes (from the first date in a data base until 30.06.2017). The major part of the retu rns of the analyzed indexes differs from the normal distribution, which question the possi bility of unreflective implementation in practice of economic such models as CAPM and its derivatives, Black–Scholes options valuation, portfolio theory and efficient market hypothesis, especially in long time horizons. Contribution/value/contribution – The contribution of this paper is verification of the statistical hypothesis regarding normal dist ribution of rates of return: (1) other than close-close, i.e. open-open, open-close and ove rnight with the use of various statistical tests, various data compression (daily, w eekly, monthly, quarterly, yearly) for 65 in- dexes, (2) for six stock exchange indexes in each of the years from the period of 2013- 2016 (daily data) and (3) for individual up and down waves for the DJIA index (daily data). In addition, other papers focused only on one or two statistical tests, while five different tests were implemented in this paper. This paper is the first to create a ranking of stock market indexes due to the normal distribution.






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  • Institute of Risk and Financial Market. Warsaw School of Economics, Warsaw, Poland


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